Geometry & Topology, Vol. 6 (2002)
Paper no. 8, pages 219--267.
Deformation and rigidity of simplicial group actions on trees
Max Forester
Abstract.
We study a notion of deformation for simplicial trees with group
actions (G-trees). Here G is a fixed, arbitrary group. Two G-trees are
related by a deformation if there is a finite sequence of collapse and
expansion moves joining them. We show that this relation on the set of
G-trees has several characterizations, in terms of dynamics, coarse
geometry, and length functions. Next we study the deformation space of
a fixed G-tree X. We show that if X is `strongly slide-free' then it
is the unique reduced tree in its deformation space.
These methods allow us to extend the rigidity theorem of Bass and
Lubotzky to trees that are not locally finite. This yields a unique
factorization theorem for certain graphs of groups. We apply the
theory to generalized Baumslag-Solitar groups and show that many have
canonical decompositions. We also prove a quasi-isometric rigidity
theorem for strongly slide-free G-trees.
Keywords.
G-tree, graph of groups, folding, Baumslag-Solitar group, quasi-isometry
AMS subject classification.
Primary: 20E08.
Secondary: 57M07, 20F65.
DOI: 10.2140/gt.2002.6.219
E-print: arXiv:math.GR/0107008
Submitted to GT on 21 June 2001.
(Revised 21 March 2002.)
Paper accepted 3 May 2002.
Paper published 3 May 2002.
Notes on file formats
Max Forester
Mathematics Institute, University of Warwick
Coventry, CV4 7AL, UK
Email: forester@maths.warwick.ac.uk
GT home page
