Quasi-actions on trees I. Bounded valence

By Lee Mosher, Michah Sageev, K. Whyte. 2000

111 citations recorded by the source corpus.

Given a bounded valence, bushy tree T, we prove that any cobounded quasi-action of a group G on T is quasiconjugate to an action of G on another bounded valence, bushy tree T'. This theorem has many applications: quasi-isometric rigidity for fundamental groups of finite, bushy graphs of coarse PD(n) groups for each fixed n; a generalization to actions on Cantor sets of Sullivan's theorem about uniformly quasiconformal actions on the 2-sphere; and a characterization of locally compact topological groups which contain a virtually free group as a cocompact lattice. Finally, we give the first examples of two finitely generated groups which are quasi-isometric and yet which cannot act on the same proper geodesic metric space, properly discontinuously and cocompactly by isometries.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/38a08feed85d1cae10737ff0b1a0c234a110728c

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Harmonic analysis, cohomology, and the large-scale geometry of amenable groups (2004, background): “[43] where it is called "uniformly proper embedding" or [11] where it is termed "packing", and the references therein).”
Coarse differentiation of quasi-isometries I: Spaces not quasi-isometric to Cayley graphs (2006, methodology): “s that there is a dwith d1 = du, d2 = dv for some uand v. There is still some ambiguity in the choices, since many groups occur as subgroups of Sim(Qpk) for many diﬀerent k. As in the construction of [MSW] we can make the choices unique by choosing theti themaximum possible, sothatallpowers ofdi t i”
Geometry of pseudocharacters (2003, unknown): “We use ‖δf‖ to denote the smallest nonnegative C so that |δf(α, β)| ≤ C for all α, β in G. Geometry & Topology, Volume 9 (2005) We fix a group G and a pseudocharacter f : G → R.”
The Word and Geodesic Problems in Free Solvable Groups (2008, result): “We refer to the papers [ 41 ] and [17] for some recent results in this area.”
Analogues of Cayley graphs for topological groups (2008, background): “The above Corollary implies a special case of [ 43, Theorem 9 ].”
Quasi-Actions on Trees II: Finite Depth Bass-Serre Trees (2004, methodology): “eorem of [FS96], and the more general coarse Jordan separation theorem and coarse Alexander duality theorem of [KK99]. The material of this section is taken from [KK99], as extended to CWcomplexes in [MSW03]. 22 Bounded geometry CW-complexes. Let Y be a ﬁnite dimensional CW-complex. Using induction ”
The asymptotic geometry of right-angled Artin groups, I (2007, background): “ly on some tree [Sta68], [Dun85], [Gro93, 1.C1] and is commensurable to afree group[KPS73]. Furthermore, any quasi-action G′ y K is quasi-isometrically conjugate to an isometric action on a tree, see [MSW03] and Section 2. • G= Fk×Fl. The model space Kis a product of simplicial trees; as with free g”
Constructing group actions on quasi-trees and applications to mapping class groups (2010, background): “mplex Pd(X) for the Cayley graph Xof the group and large d. Theorem G is not a counterexample to this approach since our Xis not locally ﬁnite, but it does point out diﬃculties. Note that in light of [MSW03] the quasi-trees that arise in our construction are necessarily locally inﬁnite, since otherw”
On the automorphism group of generalized Baumslag–Solitar groups (2005, background): “Any torsion-free group quasi-isometric to Fn × Z with n > 1 is a unimodular GBS g roup [26].”