Modulii of solenoids and commensurations of the free group

Abstract: The abstract commensurator of a group Comm(G) is the group of isomorphisms between finite-index subgroups, up to the equivalence of agreement on a common finite-index domain. Commensurators famously appear in Margulis' arithmeticity theorem, and for a suitable Lie group the abstract commensurator detects the arithmeticity of lattices. Notably, PSL_2(R) is not suitable: every lattice is virtually a free or closed surface group, and the abstract commensurator fails to detect arithmeticity in this setting. Other methods are required to study the abstract commensurator of finitely generated free or surface groups. Biswas, Nag, and Sullivan developed a universal Teichmuller theory" and approached the abstract commensurator of a surface group with ideas from mapping class group theory. I will introduce, following Culler and Vogtmann's outer space, the modulii space of metric graph solenoids. This space is equivalent to an appropriate direct limit of auter spaces (base points turn out important), and has a global structure coming from Leighton's theorem on graph covers. Once the audience is situated in this space I will use the natural action of the abstract commensurator Comm(F) on this space to reveal exciting features of Comm(F). This is joint work with Daniel Studenmund.

algebraic topologydifferential geometrydynamical systemsgroup theorygeometric topologysymplectic geometry

Audience: researchers in the topic

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.