Quasi-isometries between groups with infinitely many ends

Papasoglu, Panos; Whyte, Kevin

doi:10.1007/s00014-002-8334-2

Cited by 54 publications

(74 citation statements)

References 8 publications

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“…As pointed out in [44,Theorem 0.4], the property of a locally finite connected transitive graphs being accessible is preserved by quasi-isometries.…”

Section: Definition 9 ([53 P 249])mentioning

confidence: 98%

Analogues of Cayley graphs for topological groups

Krön

Möller

2007

Math. Z.

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ABSTRACT. We define for a compactly generated totally disconnected locally compact group a graph, called a rough Cayley graph, that is a quasi-isometry invariant of the group. This graph carries information about the group structure in an analogue way as the ordinary Cayley graph for a finitely generated group. With this construction the machinery of geometric group theory can be applied to topological groups. This is illustrated by a study of groups where the rough Cayley graph has more than one end and a study of groups where the rough Cayley graph has polynomial growth.

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“…As pointed out in [44,Theorem 0.4], the property of a locally finite connected transitive graphs being accessible is preserved by quasi-isometries.…”

Section: Definition 9 ([53 P 249])mentioning

confidence: 98%

Analogues of Cayley graphs for topological groups

Krön

Möller

2007

Math. Z.

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“…That (3) implies (1) follows from the easy fact that Z n is quasi-isometric to Z m if and only if n = m plus the result of Papasoglu and Whyte [4]: a pair of groups with infinitely many ends which admit graph of groups decompositions with finite edge groups are quasi-isometric if and only if the groups both have the same set of quasi-isometry types of vertex groups.…”

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confidence: 94%

Commensurability and QI classification of free products of finitely generated abelian groups

Behrstock

Januszkiewicz²,

Neumann

2008

Proc. Amer. Math. Soc.

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The following gives the complete commensurability and quasi-isometry classification of free products of finitely generated abelian groups. The quasi-isometry classification is a special case of Papasoglu and Whyte [4]. Theorem 1. Let G i be a free product of a finite set S i of finitely generated abelian groups for i = 1, 2. Then the following are equivalent(1) The sets of ranks ≥ 2 of groups in S 1 and S 2 are equal (the rank of a finitely generated group is the rank of its free abelian part); (2) G 1 and G 2 are commensurable. (3) G 1 and G 2 are quasi-isometric.Proof. The main step is to show (1) implies (2). By going to finite index subgroups of G 1 and G 2 we can assume the groups in S 1 and S 2 are free abelian (take the kernel of the map of G i to a product of finite quotients of the groups in S i by torsion free normal subgroups, or see the lemma below for a more general statement). Let n 1 = 1 and let n 2 , . . . , n k be the ranks ≥ 2 of the groups in S 1 and in S 2 . Let r i and s i be the number of rank n i groups in S 1 and S 2 respectively. We identify G 1 with the fundamental group of the topological space W 1 consisting of the wedge of r i n i -dimensional tori for each i; similarly we let G 2 = π 1 (W 2 ), where W 2 is defined similarly using the s i 's. A finite cover of such a wedge of tori is homotopy equivalent to a wedge of tori.We proceed in two steps. First, using finite covers we replace (r 1 , r 2 , . . . , r k ) and (s 1 , s 2 , . . . , s k ) by the sequences (R 1 , Y, Y, . . . , Y ) and (S 1 , Y, Y, . . . , Y ), respectively, where R 1 , S 1 , and Y are positive integers. Then we show that, again taking finite covers, we can leave the Y 's unchanged and replace both R 1 and S 1 by a positive integer X, making the two sequences equal and completing the argument.For the first step, let Y be a common multiple of r 2 , . . . , r n , s 2 , . . . s n . We construct W 1 and W 2 in the following way. W 1 is a finite cover of W 1 which for each 2 ≤ i ≤ k satisfies the following: it has Y tori of dimension n i ; each n i -dimensional torus in W 1 has Y /r i n i -dimensional tori in W 1 which project to it; and each torus of W 1 covers its image in W 1 with degree r i . Hence W 1 is a covering of W 1 with degree Y . Similarly construct W 2 from W 2 using the s i . Note that W 1 and W 2 are each homotopy equivalent to wedges of tori (by contracting an embedded tree connecting the lifts of the basepoint), but this construction doesn't control the number of 1-dimensional tori in these wedges of tori.For step 2 we notice that, given two spaces which are homotopy equivalent to wedges of tori of dimension up to n, if the number of n i -dimensional tori is the same in each for all 2 ≤ i ≤ k, then the number of 1-tori in the equivalent wedges

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“…In particular, the groups G n are pairwise non-quasi-isometric. Notice that without the 1-ended assumption, the result easily follows from the existence of an infinite family of hyperbolic groups of unbounded asymptotic dimension and [PW02] by considering free products.…”

Section: Introductionmentioning

confidence: 99%

Relatively hyperbolic groups with fixed peripherals

Cordes

Hume

2019

Isr. J. Math.

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We build quasi-isometry invariants of relatively hyperbolic groups which detect the hyperbolic parts of the group; these are variations of the stable dimension constructions previously introduced by the authors.We prove that, given any finite collection of finitely generated groups H each of which either has finite stable dimension or is non-relatively hyperbolic, there exist infinitely many quasi-isometry types of one-ended groups which are hyperbolic relative to H.The groups are constructed using small cancellation theory over free products.

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Quasi-isometries between groups with infinitely many ends

Cited by 54 publications

References 8 publications

Analogues of Cayley graphs for topological groups

Analogues of Cayley graphs for topological groups

Commensurability and QI classification of free products of finitely generated abelian groups

Relatively hyperbolic groups with fixed peripherals

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