Boundaries/Section1 - Problems in Geometric Group Theory

1 Background

Ideal boundaries of hyperbolic spaces Suppose that $X$ is a hyperbolic metric space. Pick a base-point $o\in X$ . This defines the Gromov product $(x,y)_o\in \R_+$ for points $x, y\in X$ . The ideal boundary $\geo X$ of $X$ is the collection of equivalence classes $[x_i]$ of sequences $(x_i)$ in $X$ where $(x_i)\sim (y_i)$ if and only if

$\lim_{i\to\infty} (x_i, y_i)_o =\infty.$

The topology on $\geo X$ is defined as follows. Let $\xi\in \geo X$ . Define $r$ -neighborhood of $\xi$ to be

$U(\xi,r):= \{\eta\in \geo X: \exists (x_i), (y_i) \hbox{~~with~~} \xi=[x_i], \eta=[y_i], \liminf_{i, j\to \infty} (x_i, y_j)_o\ge r\}.$

Then the basis of topology at $\xi$ consists of $\{U(\xi,r), r\ge 0\}$ . We will refer to the resulting ideal boundary $\geo X$ as the Gromov--boundary of $X$ . One can check that the topology on $\geo X$ is independent of the choice of the base-point. Moreover, if $f: X\to Y$ is a quasi-isometry then it induces a homeomorphism $\geo f: \geo X\to \geo Y$ . The Gromov product extends to a continuous function

$(\xi, \eta)_o: \geo X\times \geo X\to [0, \infty].$

The geodesic boundary of $X$ admits a family of visual metrics $d_{\infty}^a$ defined as follows. Pick a positive parameter $a$ . Given points $\xi, \eta\in \geo X$ consider various chains $c=(\xi_1,...,\xi_m)$ (where $m$ varies) so that $\xi_1=\xi, \xi_m=\eta$ . Given such a chain, define

$d_c(\xi,\eta):= \sum_{i=1}^{m-1} e^{-a(\xi_i, \xi_{i+1})_o},$

where $e^{-\infty}:=0$ . Finally,

$d^a_\infty(x,y):= \inf_{c} d_c(\xi,\eta)$

where the infimum is taken over all chains connecting $\xi$ and $\eta$ . Taking different values of $a$ results in H\"older--equivalent metrics. Each quasi-isometry $X\to Y$ yields a quasi-symmetric homeomorphism (see section \ref{analytical} for the definition)

$(\geo X, d_{\infty}^a)\to (\geo Y, d_{\infty}^a).$

Conversely, each quasi-symmetric homeomorphism as above extends to a quasi-iso\-metry $X\to Y$ , see MR1395067.

Then the ideal boundary of a Gromov--hyperbolic group $G$ is defined as

$\geo \Ga_G,$

where $\Ga_G$ is a Cayley graph of $G$ . Hence $\Ga_G$ is well-defined up to a quasi-symmetric homeomorphism.

Ideal boundaries of spaces Consider a $CAT(0)$ space $X$ . Two geodesic rays $\al, \be: \R_+\to X$ are said to be equivalent if there exists a constant $C\in \R$ such that

$d(\al(t), \be(t))\le C, \forall t\in \R_+.$

The geodesic boundary $\geo X$ of $X$ is defined to be the set of equivalence classes $[\al]$ of geodesic rays $\al$ in $X$ . Fix a base-point $o\in X$ . If $X$ is locally compact (which we will assume from now on), then there exists a unique a representative $\al$ in each equivalence class $[\al]$ so that $\al(0)=o$ . With this convention the visual topology on $\geo X$ is defined as the compact-open topology on the space of maps $\R_+\to X$ . One can check that this topology is independent of the choice of the base-point and that isometries $X\to Y$ induce homeomorphisms $\geo X\to \geo Y$ .

Example. If $X=\R^n$ then $\geo X$ is homeomorphic to $S^{n-1}$ .

If $X$ is a $CAT(-1)$ space then it is also Gromov-hyperbolic. Then the two ideal boundaries of $X$ (one defined via sequences and the other defined via geodesic rays) are canonically homeomorphic to each other. More specifically, each geodesic ray $\al$ defines sequences $x_i=\al(t_i)$ , for $t_i\in \R_+$ diverging to infinity. The equivalence class of such $(x_i)$ is independent of $(t_i)$ and one gets a homeomorphism from the $CAT(0)$ -boundary to the Gromov-boundary.

In general, quasi-isometries of $CAT(0)$ spaces to not extend to the ideal boundaries in any sense. Moreover, Bruce Kleiner and Chris Croke constructed examples MR1746908 of pairs of spaces $X, X'$ which admit geometric (i.e. isometric, discrete, cocompact) actions by the same group $G$ so that $\geo X, \geo X'$ are not homeomorphic.

Therefore, given a $CAT(0)$ --group $G$ one can talk only of the collection of boundaries of $G$ , i.e. the set

$\{ \geo X: \exists G\acts X\}$

where the actions $G\acts X$ are geometric.

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