The Farrell--Tate K-theory of $\mathrm{Out}(F_n)$

Abstract: Given a (nice enough) group, there is an isomorphism, due to Lück, relating the rationalised K-theory groups of its classifying space to a large product of cohomology groups, some with rational and some with $p$-adic coefficients.

We identify a generalised cohomology theory capturing the $p$-adic part of this product. Working in $\mathrm{Out}(F_n)$, in ranks close to $p$ we can fully compute this $p$-adic part and in this way produce an infinite family of odd-dimensional summands in the rationalised K-theory of $\mathrm{Out}(F_n)$. I will discuss these results and the tools that go into them, which range from spherical group rings to the lemma that is not Burnside's, via results about centralisers in $\mathrm{Out}(F_n)$: I will try to explain how all these various ideas fit together!

This is joint work with Irakli Patchkoria.

algebraic topologydifferential geometrydynamical systemsgroup theorygeometric topologysymplectic geometry

Audience: researchers in the topic

( paper )

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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.

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