Pure maths colloquium - Richard Webb
| Dates: | 11 November 2026 |
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| Times: | 13:30 - 14:30 |
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| What is it: | Seminar |
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| Organiser: | Department of Mathematics |
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| Who is it for: | University staff, External researchers, Current University students |
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Speaker: Richard Webb (University of Manchester)
Title: Finding surface subgroups in mapping class groups
Abstract: Many infinite groups can be studied by associating geometric spaces to them. Free groups, for example, act naturally on trees. The class of hyperbolic groups generalises this picture, in that they are modelled on spaces where geodesic triangles are uniformly slim, a fact that is true for trees. I will explain a few ways in which their geometry helps us understand their algebra. These groups play a central role in low-dimensional topology and geometric group theory. Beyond free groups, basic examples are the surface groups: the fundamental groups of closed orientable surfaces of genus at least two. These groups act on the hyperbolic plane.
While many infinite groups contain an abundance of free subgroups, finding surface subgroups in general is notoriously difficult. In this talk, I will focus on mapping class groups, whose elements are homeomorphisms of a surface to itself up to continuous deformation. Although mapping class groups are generally not hyperbolic, they exhibit many features in common with hyperbolic groups. I will explain how to find surface subgroups of mapping class groups with desirable geometric properties. Our construction also produces surface bundles over surfaces whose total spaces have hyperbolic fundamental groups. Joint work with Alex Wright.
Room: Alan Turing Building G.205
The talk will be followed by a reception outside of FA1.
Travel and Contact Information
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Frank Adams 1
Alan Turing Building
Manchester
