Quasi-isometries between groups with infinitely many ends
نویسندگان
چکیده
Let G, F be finitely generated groups with infinitely many ends and let π1(Γ,A), π1(∆,B) be graph of groups decompositions of F, G such that all edge groups are finite and all vertex groups have at most one end. We show that G, F are quasi-isometric if and only if every one-ended vertex group of π1(Γ,A) is quasi-isometric to some one-ended vertex group of π1(∆,B) and every one-ended vertex group of π1(∆,B) is quasi-isometric to some one-ended vertex group of π1(Γ,A). From our proof it also follows that if G is any finitely generated group, of order at least three, the groups: G ∗G, G ∗ Z, G ∗G ∗G and G ∗ Z/2Z are all quasi-isometric. Mathematics Subject Classification (2000). 20F65, 20E06.
منابع مشابه
Group Splittings and Asymptotic Topology
It is a consequence of the theorem of Stallings on groups with many ends that splittings over finite groups are preserved by quasi-isometries. In this paper we use asymptotic topology to show that group splittings are preserved by quasi-isometries in many cases. Roughly speaking we show that splittings are preserved under quasi-isometries when the vertex groups are fundamental groups of aspheri...
متن کاملar X iv : m at h / 02 01 31 2 v 1 [ m at h . G R ] 3 1 Ja n 20 02 GROUP SPLITTINGS AND ASYMPTOTIC TOPOLOGY
It is a consequence of the theorem of Stallings on groups with many ends that splittings over finite groups are preserved by quasi-isometries. In this paper we use asymptotic topology to show that group splittings are preserved by quasi-isometries in many cases. Roughly speaking we show that splittings are preserved under quasi-isometries when the vertex groups are fundamental groups of aspheri...
متن کاملQuasi-isometries between graphs and trees
Criteria for quasi-isometry between trees and general graphs as well as for quasi-isometries between metrically almost transitive graphs and trees are found. Thereby we use different concepts of thickness for graphs, ends and end spaces. A metrically almost transitive graph is quasi-isometric to a tree if and only if it has only thin metric ends (in the sense of Definition 3.6). If a graph is q...
متن کاملRigidity of quasi-isometries for symmetric spaces and Euclidean buildings
for all x ∈ X . Quasi-isometries occur naturally in the study of the geometry of discrete groups since the length spaces on which a given finitely generated group acts cocompactly and properly discontinuously by isometries are quasi-isometric to one another [Gro]. Quasi-isometries also play a crucial role in Mostow’s proof of his rigidity theorem: the theorem is proved by showing that equivaria...
متن کاملRigidity and Equivalence Relations with Infinitely Many Ends
We consider groups and equivalence relations with infinitely many ends and the problem of selecting one end in a uniform manner. In general a non-amenable equivalence relation may have infinitely many ends and yet admit a Borel function selecting from each class a single end; however, we show that in the presence of an invariant Borel probability measure, the equivalence having infinitely many ...
متن کامل