Quasi-isometries between groups with infinitely many ends

By Panos Papazoglu, K. Whyte. 2002

92 citations recorded by the source corpus.

Abstract. Let G, F be finitely generated groups with infinitely many ends and let¶ $ \pi_1(\Gamma,\mathcal A), \pi_1(\Delta ,\mathcal 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 $ \pi_1(\Gamma, \mathcal A) $ is quasi-isometric to some one-ended vertex group of $ \pi_1(\Delta, \mathcal B) $ and every one-ended vertex group of $ \pi_1(\Delta, \mathcal B) $ is quasi-isometric to some one-ended vertex group of¶ $ \pi_1(\Gamma, \mathcal A) $. From our proof it also follows that if G is any finitely generated group, of order at least three, the groups: $ G \ast G, G \ast \mathbb{Z}, G \ast G \ast G $ and $ G \ast \mathbb{Z}/2\mathbb{Z} $ are all quasi-isometric.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/18d0eb47128e50931f948c45cd1d4a6c14181ef9

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

Citations in the corpus

Quasi-Actions on Trees II: Finite Depth Bass-Serre Trees (2004, result): “The first example of this type is Stallings Ends Theorem, which implies that if G splits with finite edge groups, then so must H. This result was recently refined in work of Papasoglu and Whyte [ PW02 ] which implies that if G and H are accessible groups then they have the same quasi-isometry types ”
Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity (2005, background): “By results in [76], the classification of relatively hyperbolic groups reduces to the classification of one-ended relatively hyperbolic groups.”
Tree-graded spaces and asymptotic cones of groups (2004, methodology): “A result similar to Theorem 1.7 is obtained in [ PW , x3] for G a fundamental group of a graph of groups with flnite edge groups and S a one-ended group.”