Abstract

Ulam stability asks whether approximate homomorphisms are close to homomorphisms. It is much studied in the context of homomorphisms into groups of linear operators. We shall discuss it in the context of maps from a finitely generated group G to Sym(n), as n tends to infinity, using the normalized Hamming metric d_n on Sym(n). More precisely, we fix a group G, and ask whether for any given sequence {f_n} of maps f_n : G → Sym(n), such that d_n(f_n(xy),f_n(x)f_n(y)) → 0 for all x,y in G, there is a sequence {h_n} of homomorphisms h_n : G → Sym(n) such that d_n(f_n(x),h_n(x)) → 0 for all x in G. This question has a pointwise version and a uniform version, and we will discuss both. In any case, the answer depends on the group G.

The study of stability involves notions such as amenability, invariant random subgroups, property (T), sofic groups and basis reduction theory. We shall survey known results and the connections to property testing and to approximability in group theory.

The talk is based on joint works with Alex Lubotzky, Andreas Thom, Jonathan Mosehiff and Michael Chapman.

More details will be given in the course Approximate Group Actions and Ulam Stability, given on Mondays and Wednesdays throughout Lent term.