On the quasi-isometric rigidity of graphs of surface groups

Abstract

Let $\Gamma$ be a word hyperbolic group with a cyclic JSJ decomposition that has only rigid vertex groups, which are all fundamental groups of closed surface groups. We show that any group $H$ quasi-isometric to $\Gamma$ is abstractly commensurable with $\Gamma$.

After some revisions, the referee identified a gap in the proof of Lemma 4.20. The following results are impacted: Corollary 4.28, Corollary 5.19, and Theorem A. The authors believe that Corollary 4.28 may be false. What can be recovered from Sections 5 and 6 is that Churro-Waffle spaces are action rigid. The authors still believe that Theorem A is true and are currently working on another proof

References in corpus (3)

• JSJ decompositions of groups
• Quasi-isometries Between Groups with Two-Ended Splittings
• Quasi-isometric groups with no common model geometry

Cited by in corpus (1)

• Quasi-isometric rigidity for graphs of virtually free groups with two-ended edge groups