18.108 (2014)

A group $G$ is said to have property $(\tau)$ if its trivial representation is an isolated point in the subspace of irreducible representations with finite images (in the topological unitary dual space of $G$). Is it true that there exists a finitely generated group satisfying property $(\tau)$ that homomorphically maps onto every finite simple group?

The motivation is the theorem in (E. Breuillard, B. Green, M. Kassabov, A. Lubotzky, N. Nikolov, T. Tao) saying that there are $\epsilon > 0$ and $k$ such that every finite simple group $G$ contains a generating set $S$ with at most $k$ elements such that the Cayley graph of $G$ with respect to $S$ is an $\epsilon$-expander. In (M. Ershov, A. Jaikin Zapirain, M. Kassabov, Mem. Amer. Math. Soc., 1186, 2017) it is proved that there exists a group with Kazhdan’s property (T) that maps onto every simple group of Lie type of rank $\geqslant 2$. (A group $G$ is said to have Kazhdan’s property (T) if its trivial representation is an isolated point in the unitary dual space of $G$.)