8 Analytical aspects of boundaries of groups
- \label{analytical}
We begin with the basic definitions of the quasiconformal analysis. For a quadruple of points
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Quasiconformal (analytic definition): A homeomorphism
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The essential supremum
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. Note: The map
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is differentiable almost everywhere, so the derivative and Jacobian in (2) make sense pointwise a.e.. The assumption (1) can be replaced by the assumption that latex error! exitcode was 1 (signal 0), transscript follows:
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! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_37eea899b7b5618e7d822164f0c7b70f36a52bc9_p.log.
latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_65cf542047f4dbe238a0acd564df1b684191b06a_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_65cf542047f4dbe238a0acd564df1b684191b06a_p.log.
latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_baba46dc4404ca3da0ec3065aa49269ed7d3afb0_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_baba46dc4404ca3da0ec3065aa49269ed7d3afb0_p.log.
Then,
latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_ddc66ff99a78e4a7f75459d39bf1ae66928851e4_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_ddc66ff99a78e4a7f75459d39bf1ae66928851e4_p.log.
is quasiconformal iff latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_afc5038163e82522ab0b3caa3961294f4af7c062_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_afc5038163e82522ab0b3caa3961294f4af7c062_p.log.
is uniformly bounded by some latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_ffb9cad64c395032960cfbf8b6b95e47625639fa_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_ffb9cad64c395032960cfbf8b6b95e47625639fa_p.log.
. 2. Quasiconformal (geometric definition): Let
latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_ceb63fe3297d1b4af9704e31f6a6d2aecc37edc8_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_ceb63fe3297d1b4af9704e31f6a6d2aecc37edc8_p.log.
be a family of paths in latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_c87995b89e132ff6840a4a9fc393e05158c4484e_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_c87995b89e132ff6840a4a9fc393e05158c4484e_p.log.
. We say a Borel function latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_70d4f94f684e5e2a4230578850098dfb14cd3708_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_70d4f94f684e5e2a4230578850098dfb14cd3708_p.log.
is admissible for latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_ceb63fe3297d1b4af9704e31f6a6d2aecc37edc8_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_ceb63fe3297d1b4af9704e31f6a6d2aecc37edc8_p.log.
iff for every latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_47bdd65db41aa8b3764eca9a703419cf1fe7aa26_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_47bdd65db41aa8b3764eca9a703419cf1fe7aa26_p.log.
, we have latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_ee383130dfc26ccd4154d234f2f20b48de044dc7_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_ee383130dfc26ccd4154d234f2f20b48de044dc7_p.log.
. Define
latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_7a2296b2e1043c11a359218d6fadb39952142595_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_7a2296b2e1043c11a359218d6fadb39952142595_p.log.
. A homeomorphism latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_510a8561a055ac03ec687d373ad524abc3c172e7_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_510a8561a055ac03ec687d373ad524abc3c172e7_p.log.
is quasiconformal iff there is some constant latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_e5f887ad02162d7913bda519c983cfb1c97d0eb4_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_e5f887ad02162d7913bda519c983cfb1c97d0eb4_p.log.
such that for every path family latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_ceb63fe3297d1b4af9704e31f6a6d2aecc37edc8_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_ceb63fe3297d1b4af9704e31f6a6d2aecc37edc8_p.log.
, we have latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_848c2d0169e16da7857b1561e601a3e2cd24789b_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_848c2d0169e16da7857b1561e601a3e2cd24789b_p.log.
3. A homeomorphism
latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_22779ff693d7403499b305f38a24b9fcaff18922_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_22779ff693d7403499b305f38a24b9fcaff18922_p.log.
between metric spaces latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_b03e95f5b8ab543a0d7b2a79a392f1f5557a3e33_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_b03e95f5b8ab543a0d7b2a79a392f1f5557a3e33_p.log.
and latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_ddbebb3fd2cdc3c6f97b5065b3a2a1b07d068e90_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_ddbebb3fd2cdc3c6f97b5065b3a2a1b07d068e90_p.log.
is quasisymmetric iff there exists a homeomorphism latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_dc476960d2dc2a7abffbfc1fc73161b88dd12af9_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_dc476960d2dc2a7abffbfc1fc73161b88dd12af9_p.log.
such that for all triples of distinct points latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_4a8da09c171a6cb9db24d97bb6f19f900953bab3_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_4a8da09c171a6cb9db24d97bb6f19f900953bab3_p.log.
, the following inequality holds: latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_4c73bd2fbeaf118ac9bc9749dabed1130bce7ef4_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_4c73bd2fbeaf118ac9bc9749dabed1130bce7ef4_p.log.
4. A homeomorphism
latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_22779ff693d7403499b305f38a24b9fcaff18922_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_22779ff693d7403499b305f38a24b9fcaff18922_p.log.
between metric spaces latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_b03e95f5b8ab543a0d7b2a79a392f1f5557a3e33_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_b03e95f5b8ab543a0d7b2a79a392f1f5557a3e33_p.log.
and latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_ddbebb3fd2cdc3c6f97b5065b3a2a1b07d068e90_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_ddbebb3fd2cdc3c6f97b5065b3a2a1b07d068e90_p.log.
is quasi-Moebius iff there exists a homeomorphism latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_dc476960d2dc2a7abffbfc1fc73161b88dd12af9_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_dc476960d2dc2a7abffbfc1fc73161b88dd12af9_p.log.
such that for all quadruples of distinct points latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_7ae1ed7b4e1c13878018edd7ca6ff4c757a3aade_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_7ae1ed7b4e1c13878018edd7ca6ff4c757a3aade_p.log.
, the following inequality holds: latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_284249db889811a6d53c4a4cb3db753de7ef6d9a_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_284249db889811a6d53c4a4cb3db753de7ef6d9a_p.log.
Note that Definitions 1--4 make sense in the context of general metric spaces, see below for details. If
latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_b03e95f5b8ab543a0d7b2a79a392f1f5557a3e33_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_b03e95f5b8ab543a0d7b2a79a392f1f5557a3e33_p.log.
is noncompact then quasi-symmetric maps are the same as quasi-Moebius maps. However, for compact metric spaces quasi-Moebius is a more appropriate (although more cumbersome) definition. One can rectify this problem by redefining quasi-symmetric maps for compact metric spaces as follows. A map latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_db646c275418a445c8caf1c92d2c9a37f59b37de_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_db646c275418a445c8caf1c92d2c9a37f59b37de_p.log.
is quasi-symmetric if latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_b03e95f5b8ab543a0d7b2a79a392f1f5557a3e33_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_b03e95f5b8ab543a0d7b2a79a392f1f5557a3e33_p.log.
admits a finite covering by open spaces latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_3000d0185906d7c74d8cd02e27a246c9398dff58_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_3000d0185906d7c74d8cd02e27a246c9398dff58_p.log.
so that the restriction latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_5e3fb44db5c70603889a1194562f89f53ea1baf7_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_5e3fb44db5c70603889a1194562f89f53ea1baf7_p.log.
is quasi-symmetric in the above sense for each latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_e13cc61ff14fdc2c6c7f6193a69acae222ef44b1_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_e13cc61ff14fdc2c6c7f6193a69acae222ef44b1_p.log.
. One defines a quasi-symmetric equivalence for metric spaces by
latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_f51a3eeddb4e5f0cbfd71b78ac2e890490c59c64_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_f51a3eeddb4e5f0cbfd71b78ac2e890490c59c64_p.log.
if there exists a quasisymmetric homeomorphism
latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_0d63710f47aa685396ba6368a71ecbf0c97e7194_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_0d63710f47aa685396ba6368a71ecbf0c97e7194_p.log.
. Let
latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_50d4138ef321e0794dfa4e03895c28be01017186_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_50d4138ef321e0794dfa4e03895c28be01017186_p.log.
be a metric measure space, where latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_f7b36300609b98f3c2491db9dc8fa5c32cd51d52_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_f7b36300609b98f3c2491db9dc8fa5c32cd51d52_p.log.
is a Borel measure. Then latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_b03e95f5b8ab543a0d7b2a79a392f1f5557a3e33_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_b03e95f5b8ab543a0d7b2a79a392f1f5557a3e33_p.log.
is called Ahlfors latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_2cfa6d17d4957ca38c74aee8e80e753e72056baa_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_2cfa6d17d4957ca38c74aee8e80e753e72056baa_p.log.
-regular, if there exists latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_ec3c8f546a62493a3cfee5f1bc7f3209cc4c18be_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_ec3c8f546a62493a3cfee5f1bc7f3209cc4c18be_p.log.
so that latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_e99341fea8351a307f85ed00adfaf9a7ed11543a_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_e99341fea8351a307f85ed00adfaf9a7ed11543a_p.log.
for each
latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_1354753dd025d952178234ac498a1e690bad27d4_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_1354753dd025d952178234ac498a1e690bad27d4_p.log.
. Remark Let
latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_b03e95f5b8ab543a0d7b2a79a392f1f5557a3e33_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_b03e95f5b8ab543a0d7b2a79a392f1f5557a3e33_p.log.
be a metric space with the Hausdorff dimension latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_4e2c75f6a689c4045ec17519a3174aff34c4e08d_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_4e2c75f6a689c4045ec17519a3174aff34c4e08d_p.log.
. Then the most natural measure to use is the latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_2cfa6d17d4957ca38c74aee8e80e753e72056baa_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_2cfa6d17d4957ca38c74aee8e80e753e72056baa_p.log.
-Hausdorff measure on latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_b03e95f5b8ab543a0d7b2a79a392f1f5557a3e33_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_b03e95f5b8ab543a0d7b2a79a392f1f5557a3e33_p.log.
. This is the measure to be used for the boundaries of hyperbolic groups. Then latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_9a28ac98adf7785d2b4265991e52708f67155aec_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_9a28ac98adf7785d2b4265991e52708f67155aec_p.log.
is called Ahlfors regular if it is Ahlfors latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_2cfa6d17d4957ca38c74aee8e80e753e72056baa_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_2cfa6d17d4957ca38c74aee8e80e753e72056baa_p.log.
-regular with latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_62ace0f846285e1285107124f964f129dff9c2cc_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_62ace0f846285e1285107124f964f129dff9c2cc_p.log.
. Given two compact continua
latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_81d4009a27df905bba5f59f5bffe5822f653a4a7_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_81d4009a27df905bba5f59f5bffe5822f653a4a7_p.log.
in a metric space latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_b03e95f5b8ab543a0d7b2a79a392f1f5557a3e33_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_b03e95f5b8ab543a0d7b2a79a392f1f5557a3e33_p.log.
define their relative distance latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_a3f670df6f2495cb9267b57ac9dc03a27b1e3c50_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_a3f670df6f2495cb9267b57ac9dc03a27b1e3c50_p.log.
Here
latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_60c84c8dbfd3d48aa7c5e263c8a82c8098364938_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_60c84c8dbfd3d48aa7c5e263c8a82c8098364938_p.log.
. Given an Ahlfors
latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_2cfa6d17d4957ca38c74aee8e80e753e72056baa_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_2cfa6d17d4957ca38c74aee8e80e753e72056baa_p.log.
-regular metric measure space latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_b03e95f5b8ab543a0d7b2a79a392f1f5557a3e33_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_b03e95f5b8ab543a0d7b2a79a392f1f5557a3e33_p.log.
, define latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_c678ae9f5346fc2ec899b1f898b557cb439456db_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_c678ae9f5346fc2ec899b1f898b557cb439456db_p.log.
to be latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_59500f7d3bfdd0818c46258e82cdd5e191ab17dd_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_59500f7d3bfdd0818c46258e82cdd5e191ab17dd_p.log.
where
latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_4bfbf337872b23debb82ba0ca5b5caed55cc89ff_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_4bfbf337872b23debb82ba0ca5b5caed55cc89ff_p.log.
is the set of all curves in latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_b03e95f5b8ab543a0d7b2a79a392f1f5557a3e33_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_b03e95f5b8ab543a0d7b2a79a392f1f5557a3e33_p.log.
connecting latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_c4f1e2344c6fefe1792be4230860e42cc077c12b_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_c4f1e2344c6fefe1792be4230860e42cc077c12b_p.log.
to latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_a5743856b43891a8dadc105ab8c03507fe3518a5_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_a5743856b43891a8dadc105ab8c03507fe3518a5_p.log.
. Definition 8.1 A metric space
latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_b03e95f5b8ab543a0d7b2a79a392f1f5557a3e33_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_b03e95f5b8ab543a0d7b2a79a392f1f5557a3e33_p.log.
is called latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_2cfa6d17d4957ca38c74aee8e80e753e72056baa_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_2cfa6d17d4957ca38c74aee8e80e753e72056baa_p.log.
- Loewner if it satisfies the inequality latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_d3d4103044f975582e565ad9dec4d88989689886_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_d3d4103044f975582e565ad9dec4d88989689886_p.log.
for a certain function
latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_76c24e406ca547286ad4c4cf75facf1c7322d95b_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_76c24e406ca547286ad4c4cf75facf1c7322d95b_p.log.
. Problem 8.1 (Comments) [Mario Bonk] Are diffeomorphisms
latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_245eb7734709e829b6fa472ea61332065265c3f2_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_245eb7734709e829b6fa472ea61332065265c3f2_p.log.
dense in the space of all quasiconformal maps? Remark [Juha Heinonen] The answer is known to be yes" for
latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_76e0c0222a14f1601a7ab51c52acabb2ac4515ff_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_76e0c0222a14f1601a7ab51c52acabb2ac4515ff_p.log.
due to Moise's theorem MR0488059. Remark [Misha Kapovich] In fact, the answer is also known to be "yes" for quasiconformal diffeomorphisms of
latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_3cfdd7070981b0a4edc8e9b9bf69032cc2fba74d_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_3cfdd7070981b0a4edc8e9b9bf69032cc2fba74d_p.log.
, latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_307ccfe6ee2434588a6e664a75d1a6762ae87fad_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_307ccfe6ee2434588a6e664a75d1a6762ae87fad_p.log.
. This was proven by Connell MR0154289 for stable homeomorphisms of latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_3cfdd7070981b0a4edc8e9b9bf69032cc2fba74d_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_3cfdd7070981b0a4edc8e9b9bf69032cc2fba74d_p.log.
, latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
(./latex_24d3a9e04546b226faa6ef463656d4b666ae3a6b_p.tex
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2001/04/21 v1.4e Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.118 \endinput
^^M
No pages of output.
Transcript written on latex_24d3a9e04546b226faa6ef463656d4b666ae3a6b_p.log.
, improved by Bing MR0238284 to cover dimensions latex error! exitcode was 1 (signal 0), transscript follows:
This is TeX, Version 3.14159 (Web2C 7.4.5)
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. Lastly, it was shown by Kirby MR0242165 that all orientation-preserving homeomorphisms of latex error! exitcode was 1 (signal 0), transscript follows:
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are stable for latex error! exitcode was 1 (signal 0), transscript follows:
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. Note that the proof in the case of quasiconformal homeomorphisms is easier since quasi-conformal homeomorphisms are differentiable a.e. and the stable homeomorphism conjecture was known for latex error! exitcode was 1 (signal 0), transscript follows:
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prior to Kirby's work. However the problem appears to be open in the case latex error! exitcode was 1 (signal 0), transscript follows:
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. On the other hand, Kirby observed that for sufficiently large latex error! exitcode was 1 (signal 0), transscript follows:
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there are open connected subsets latex error! exitcode was 1 (signal 0), transscript follows:
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and a homeomorphism latex error! exitcode was 1 (signal 0), transscript follows:
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, which cannot be approximated by diffeomorphisms latex error! exitcode was 1 (signal 0), transscript follows:
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. The problem becomes more subtle if we require approximating diffeomorphisms to be globally quasiconformal:
Problem 8.2 (Comments) [Misha Kapovich] Let
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be a quasiconformal homeomorphism. Can latex error! exitcode was 1 (signal 0), transscript follows:
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be approximated by globally quasiconformal diffeomorphisms latex error! exitcode was 1 (signal 0), transscript follows:
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? Can this be done so that latex error! exitcode was 1 (signal 0), transscript follows:
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's are latex error! exitcode was 1 (signal 0), transscript follows:
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-quasiconformal for all latex error! exitcode was 1 (signal 0), transscript follows:
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? Note that all the maps in problem will extend quasiconformally to the closed
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-ball. Problem 8.3 (Comments) [Mario Bonk] Find good classes of spaces such that the infinitesimal metric condition (for quasiconformality) implies the local condition. (This is generally true in Loewner spaces.)
Problem 8.4 (Comments) [Kim Ruane] Outside of the boundaries of Fuchsian buildings, what boundaries have the Loewner property?
Remark Loewner spaces are good for analytic tools: have a cotangent bundle, so can "do calculus"; also, can do PDEs, etc.
Problem 8.5 (Comments) [Juha Heinonen] Let
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be a non-smoothable closed simply connected 4-manifold. Does it admit an Ahlfors 4-regular linearly locally contractible metric? This is wide open; unknown even for examples, like latex error! exitcode was 1 (signal 0), transscript follows:
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. Remark The non-smoothable closed simply-connected 4-manifolds like
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are known not to admit a quasiconformal atlas, MR1032074. In dimensions latex error! exitcode was 1 (signal 0), transscript follows:
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Sullivan MR0537749 proved that every topological manifold admits a quasiconformal atlas and, moreover, quasiconformal structure is unique. An alternative proof of Sullivan's theorem and its generalization was given by J.~Luukkainen in MR1804561, his proof avoids the construction of almost parallelizable hyperbolic manifolds; see also MR0658932. It was observed by Tom Farrell that a detailed proof of the fact that all closed hyperbolic latex error! exitcode was 1 (signal 0), transscript follows:
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-manifolds are virtually almost parallelizable (and much more) is contained in the paper by B.~Okun MR1875614. Hence in dimension
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one would ask for Ahlfors latex error! exitcode was 1 (signal 0), transscript follows:
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-regular linearly locally contractible metrics on the unresolvable homology manifolds, see MR1394965. The broad goal here is to find an analytic framework for studying exotic topological and homology manifolds. Problem 8.6 (Comments) [Uri Bader] Develop a theory for analysis on the ideal boundaries of relatively hyperbolic groups, as it is done for hyperbolic groups.
Problem 8.7 (Comments) [Bruce Kleiner] In what generality does quasiconformal imply quasisymmetric? (Specifically, of interest are self-similar spaces which are connected, without local cut points; or visual boundaries of hyperbolic groups.)
Definition. Call a metric space
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quasi-isometrically cohopfian if each quasi-isometric embedding latex error! exitcode was 1 (signal 0), transscript follows:
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is a quasi-isometry. (Examples include Poincar\'e duality groups, solvable groups, Baumslag-Solitar groups.) latex error! exitcode was 1 (signal 0), transscript follows:
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is quasisymmetrically cohopfian if every quasisymmetric embedding latex error! exitcode was 1 (signal 0), transscript follows:
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is onto. The above definition is a coarse analogue of the notion of cohopfian groups from group theory.
Fact: a hyperbolic group
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is quasi-isometrically cohopfian iff latex error! exitcode was 1 (signal 0), transscript follows:
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is quasisymmetrically cohopfian, cf. MR1395067. Problem 8.8 (Comments) [Ilia Kapovich] Take your favorite metric fractal. Is it quasisymmetrically cohopfian? What about the boundaries of hyperbolic groups?
Subproblem: What about the case of (round) Sierpinski carpets and Menger spaces which appear as boundaries of hyperbolic groups.
Background. Round Sierpinski carpets are the ones which are bounded by round circles. Such sets arise as the ideal boundaries of fundamental groups of compact hyperbolic manifolds with nonempty totally geodesic boundary. It is known that if
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are such groups which are not commensurable then their ideal boundaries are not quasisymmetric to each other. There is a similar rigidity theorem (due to Marc Bourdon and Herve Pajot MR1789183) for a certain class of Menger curves, i.e. the ones which appear as visual boundaries of 2-dimensional Fuchsian buildings. Quasisymmetric cohopfian property is open in both cases. Remark [Danny Calegari] As an example for the previous problem: the limit set
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