Deformation and rigidity of simplicial group actions on trees

By Max Forester. 2001

113 citations recorded by the source corpus.

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.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1d4a2a713b73db9f6e42dfd6293cc36685f90f98

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Citations in the corpus

Jsj Decompositions of Groups (2016, background): “esinDdominatethoseofD0. Every deformationspacedominatesthedeformationspaceofthetrivialtree,whichiscalledthe trivialdeformationspace. ItistheonlydeformationspaceinwhichGiselliptic. A tree T is reduced [For02] if no proper collapse of T lies in the same deformation spaceasT(thisisdiﬀerentfrombeingredu”
Quotients and subgroups of Baumslag–Solitar groups (2013, background): “When q or r equals ±1, this is an elementary collapse [8] which does not change G (see Figure 3).”
Deformation spaces of trees (2006, methodology): “A tree T is reduced (in the sense of Forester [16]) if no elementary collapse is possible (collapsing any edge yields a tree in a different deformation space).”
JSJ-Decompositions of finitely presented groups and complexes of groups (2005, methodology): “We note also that the question of ‘uniqueness’ of JSJ-splittings has been treated in [ Fo ].”
On the automorphism group of generalized Baumslag–Solitar groups (2005, background): “As any two GBS trees have the same elliptic subgroups, Forester’s deformation theorem [10] yields:”
Regular neighbourhoods and canonical decompositions for groups. (2001, methodology): “In [ 15 ], Forester considered two moves on graphs of groups or equivalently G-trees called a collapse move and an expansion move.”
The outer space of a free product (2005, methodology): “In [15], we prove contractibility of arbitrary deformation spaces, using a general basepoint argument (two simplicial G-trees are in the same deformation space if they have the same elliptic subgroups, see [10]).”
Splittings and automorphisms of relatively hyperbolic groups (2012, background): “A1,A2 abelian, then Out(G;H(t)) is ﬁnite if and only if, for i = 1,2, the subgroup of Ai generated by subgroups conjugate to a group in H has ﬁnite index. This is because Γ is Out(G;H(t))- invariantby[For02] (its Bass-Serretree is the uniquereduced tree in its deformation space). Twists are trivial ”
Trees of cylinders and canonical splittings (2008, background): “Having the same elliptic subgroups is an equivalence relation on the set of trees, the equivalence classes are called deformation spaces [9; 14].”