A simple criterion for non-relative hyperbolicity and one-endedness of groups

Abstract

We give a combinatorial criterion that implies both the non-strong relative hyperbolicity and the one-endedness of a finitely generated group. We use this to show that many important classes of groups do not admit a strong relatively hyperbolic group structure and have one end. Applications include surface mapping class groups, the Torelli group, (special) automorphism and outer automorphism groups of most free groups, and the three-dimensional Heisenberg group. Our final application is to Thompson's group F.

13 pages, no figures. Argument generalised to cover a wider class of groups and to deduce their one-endedness

References in corpus (4)

• Geometry of the Complex of Curves I: Hyperbolicity
• Asymptotic geometry of the mapping class group and Teichmueller space
• Relatively hyperbolic groups: Intrinsic geometry, algebraic properties, and algorithmic problems
• Asymptotic dimension of relatively hyperbolic groups

Cited by in corpus (6)

• Random walks on the mapping class group
• Quasi-isometry rigidity of groups
• Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity
• Subgroups of the mapping class group from the geometrical viewpoint
• Uniform convergence in the mapping class group
• Continuous cocycle superrigidity for coinduced actions and relative ends