Boundaries in Geometric Group Theory
June 13 to June 17, 2005
American Institute of Mathematics,
Palo Alto, California
Misha Kapovich and Bruce Kleiner
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:
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).
- 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.
Material from the workshop
A list of participants.
The workshop schedule.
A report on the workshop activities.
A list of open problems.