We study a notion of deformation for simplicial trees with group actions (Gtrees). 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. AMS Classification numbers Primary: 20E08Secondary: 57M07, 20F65
The k -dimensional Dehn (or isoperimetric) function of a group bounds the volume of efficient ball-fillings of k -spheres mapped into k -connected spaces on which the group acts properly and cocompactly; the bound is given as a function of the volume of the sphere. We advance significantly the observed range of behavior for such functions. First, to each nonnegative integer matrix P and positive rational number r , we associate a finite, aspherical 2-complex X r;P and determine the Dehn function of its fundamental group G r;P in terms of r and the Perron-Frobenius eigenvalue of P . The range of functions obtained includes ı.x/ D x s , where s 2 Q \ OE2; 1/ is arbitrary. Next, special features of the groups G r;P allow us to construct iterated multiple HNN extensions which exhibit similar isoperimetric behavior in higher dimensions. In particular, for each positive integer k and rational s > .k C 1/=k , there exists a group with k -dimensional Dehn function x s . Similar isoperimetric inequalities are obtained for fillings modeled on arbitrary manifold pairs .M; @M / in addition to .B kC1 ; S k /.20F65; 20F69, 20E06, 57M07, 57M20, 53C99 IntroductionGiven a k -connected complex or manifold one wants to identify functions that bound the volume of efficient ball-fillings for spheres mapped into that space. The purpose of this article is to advance the understanding of which functions can arise when one seeks optimal bounds in the universal cover of a compact space. Despite the geometric nature of both the problem and its solutions, our initial impetus for studying isoperimetric problems comes from algebra, more specifically the word problem for groups.The quest to understand the complexity of word problems has been at the heart of combinatorial group theory since its inception. When one attacks the word problem for a finitely presented group G directly, the most natural measure of complexity is What Brady and Bridson actually do in [3] is associate to each pair of positive integers p > q a finite aspherical 2-complex whose fundamental group G p;q has Dehn function x 2 log 2 2p=q . These complexes are obtained by attaching a pair of annuli to a torus, the attaching maps being chosen so as to ensure the existence of a family of discs in the universal cover that display a certain snowflake geometry (cf Figure 4 below). In the present article we present a more sophisticated version of the snowflake construction that yields a much larger class of isoperimetric exponents.Theorem A Let P be an irreducible nonnegative integer matrix with Perron-Frobenius eigenvalue > 1, and let r be a rational number greater than every row sum of P . Then there is a finitely presented group G r;P with Dehn function ı.x/ ' x 2 log .r / .Here, ' denotes coarse Lipschitz equivalence of functions. By taking P to be the 1 1 matrix .2 2q / and r D 2 p (for integers p > 2q ) we obtain the Dehn function ı.x/ ' x p=q and deduce the following corollary.Corollary B Q \ .2; 1/ IP. For each positive integer k one has the k -dimensional isoperimetric spe...
We study the structure of generalized Baumslag-Solitar groups from the point of view of their (usually non-unique) splittings as fundamental groups of graphs of infinite cyclic groups. We find and characterize certain decompositions of smallest complexity (fully reduced decompositions) and give a simplified proof of the existence of deformations. We also prove a finiteness theorem and solve the isomorphism problem for generalized Baumslag-Solitar groups with no non-trivial integral moduli.
Abstract.We give an example of two JSJ decompositions of a group that are not related by conjugation, conjugation of edge-inclusions, and slide moves. This answers the question of Rips and Sela stated in [RS].On the other hand we observe that any two JSJ decompositions of a group are related by an elementary deformation, and that strongly slide-free JSJ decompositions are genuinely unique. These results hold for the decompositions of Rips and Sela, Dunwoody and Sageev, and Fujiwara and Papasoglu, and also for accessible decompositions.
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