Character varieties of random groups

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• Oren Becker, Cambridge
• Wednesday 09 February 2022, 16:00-17:00
• MR13.

If you have a question about this talk, please contact Henry Wilton.

The space Hom(Γ,G) of homomorphisms from a finitely-generated group Γ to a complex semisimple algebraic group G is known as the G-representation variety of Γ. We study this space when G is fixed and Γ is a random group in the few-relators model. That is, Γ is generated by k elements subject to r random relations of length L, where k and r are fixed and L tends to infinity.

More precisely, we study the subvariety Z of Hom(Γ,G), consisting of all homomorphisms whose images are Zariski dense in G. We give an explicit formula for the dimension of Z, valid with probability tending to 1, and study the Galois action on its geometric components. In particular, we show that in the case of deficiency 1 (i.e., k-r=1), the Zariski-dense G-representations of a typical Γ enjoy Galois rigidity.

Our methods assume the Generalized Riemann Hypothesis and exploit mixing of random walks and spectral gap estimates on finite groups.

Based on a joint work with E. Breuillard and P. Varju.

This talk is part of the Differential Geometry and Topology Seminar series.

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