### Geometry of the genus two handlebody group

Geometry and Topology Seminar

19th January 2021, 2:00 pm – 3:00 pm

Zoom seminar, if interested, please email one of the organisers to gain access to the Zoom link

In this talk, we explore the geometry of the handlebody group, i.e. the mapping class group of a handlebody. This talk will include a heuristic description of hierarchically hyperbolic spaces, and using this description, we will see that the handlebody group of genus two is a hierarchically hyperbolic group (HHG). Then, by analyzing the structure of the maximal hyperbolic space associated to the handlebody group and utilizing the characterization of stable subgroups of HHGs, I will show that the stable subgroups of the genus two handlebody group are precisely those subgroups whose orbit maps are quasi-isometric embeddings into the disk graph. Lastly, we will see that various properties of the genus two handlebody group do not hold for higher genus handlebody groups.

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