# Boundaries in Geometric Group Theory

June 13 to June 17, 2005

at the

American Institute of Mathematics, Palo Alto, California

organized by

Misha Kapovich and Bruce Kleiner

## Original Announcement

This workshop will serve to facilitate discussions on various aspects of boundaries of groups. A variety of different constructions of boundaries at infinity have long played an important role in geometric group theory and related areas such as Kleinian groups, rigidity theory for lattices, 3-manifolds, and nonpositive curvature. In the last 7 years, there has been particularly intense progress on boundaries, especially boundaries of hyperbolic groups and CAT(0) groups. For example:
• Work of Bestvina-Mess, Bowditch, Swarup, Swenson and others settled the fundamental cut-point conjecture, simplified and strengthened the proof, and applied it to the structure theory of hyperbolic groups.
• Papers by Rips-Sela, Bowditch, Papasoglu, Dunwoody-Swenson, and Dunwoody-Sageev established JSJ decompositions for finitely presentable groups; these results either use boundary dynamics in their proofs, or use boundary structure to establish quasi-isometry invariance of the splittings.
• Bowditch and Swenson have shown that a group is necessarily hyperbolic provided it has an action on a compact space satisfying suitable dynamical properties. These criteria have been applied to special cases of the weak hyperbolization conjecture, and the algebraic torus theorem of Dunwoody-Swenson.
• Work of Heinonen-Koskela, Tyson, Bourdon-Pajot, Bonk-Kleiner, and Keith-Laakso significantly advanced the theory of quasiconformal and quasisymmetric homeomorphisms Poincare inequalities, and applied it to obtain rigidity and uniformization results for boundaries of hyperbolic groups.
• There a many other notable results on other types of boundaries, for instance the Poisson boundary, $G$-boundaries in the sense of Bestvina, and the Floyd boundary.
The principal goal of the workshop is to bring together people working on this topic so as to foster further research. There will be several survey lectures to help participants take stock of all the new developments, and disseminate ideas across customary barriers (expose the more analytic ideas to topologists, and topological ideas to analysts, etc).

## Material from the workshop

A list of participants.

The workshop schedule.

A report on the workshop activities.