Event № 809
The following question is well-studied: Are almost commuting matrices necessarily close to commuting matrices? Both positive and negative answers were given, depending on the types of matrices considered and the metrics used to measure proximity. Variants of the question replace the matrices by different objects and/or replace the commutativity relation by another one. This suggests the following framework:
Fix a family \calG of pairs (G,d), where G is a group and d is a bi-invariant metric on G. For example, one may take \calG to be the family of finite symmetric groups endowed with the normalized Hamming metrics, or take it to be the family of unitary groups endowed with your favorite bi-invariant metrics on the groups U(n). Fix a word w over S±, where S is a finite set of formal variables. We say that w is \calG-stable if for every \epsilon>0 there is \delta>0 such that for every (G,d) in \calG and f:S-->G, if d(f(w),1_G) <= \delta, then there is f':S-->G such that f' is \epsilon-close to f and f'(w)=1_G. The stability of a set of words, representing simultaneous equations, is defined similarly.
It turns out that the \calG-stability of a set E of words depends only on the group \Gamma generated by S subject to the relations E. In other words, stability is a group property. A finitely generated group \Gamma is \calG-stable if one (hence all) of its presentations corresponds to a \calG-stable set of words.
We will give a survey of this topic, and then focus on recent results on stability of a finitely generated group \Gamma w.r.t. symmetric groups and unitary groups. These results relate stability to notions such as amenability, Invariant Random Subgroups, Property (T) and mapping class groups.
Based on joint works with Alex Lubotzky, Andreas Thom and Jonathan Mosheiff.