### Residually finite groups with uniformly almost flat quotients

Combinatorics Seminar

29th October 2024, 11:00 am – 12:00 pm

Fry Building, 2.04

Given a residually finite group, there is considerable interest in understanding what information is encoded in the finite quotients. Khukhro and Valette investigated what algebraic properties we can deduce from the diameter of finite quotients of the group. They showed that the finite quotients have diameter bounded uniformly below by a constant multiple of their size if and only if the group is virtually cyclic, and that if the diameter of finite quotients is bounded uniformly below by a polynomial in their size then the group virtually maps onto Z. We conjecture that this latter assumption in fact implies that the group is virtually nilpotent, and we prove our conjecture or variants of it for various specific classes of groups. Joint work with Matthew Tointon.

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