Groups quasi-isometric to planar graphs

Abstract: A classic and important theorem, originating in work of Mess, states that a finitely generated group is quasi-isometric to a complete Riemannian plane if and only if it is a virtual surface group. Another related result obtained by Maillot states that a finitely generated group is virtually free if and only if it is quasi-isometric to a complete planar simply connected Riemannian surface with non-compact geodesic boundary. These results illustrate the general philosophy that planarity is a very `rigid' property amongst finitely generated groups.

In this talk I will build on the above and sketch how to characterise those finitely generated groups which are quasi-isometric to planar graphs. Such groups are virtually free products of free and surface groups, and thus virtually admit a planar Cayley graph. The main technical step is proving that such a group is accessible, in the sense of Dunwoody and Wall. This is achieved through a careful study of the dynamics of quasi-actions on planar graphs.

algebraic topologydifferential geometrydynamical systemsgroup theorygeometric topologysymplectic geometry

Audience: researchers in the topic

( paper )

---

Geometry and topology online

Series comments: You can also find up-to-date information on the seminar homepage - warwick.ac.uk/fac/sci/maths/research/events/seminars/areas/geomtop/

The talks start at 13:30. Talks are typically fifty minutes long, with ten minutes for questions.

---