Given a group G, one can study the behaviour of word maps (with constants) w : G^n \to G: these are defined by plugging in group elements into natural multiplicative expressions one can write in groups (e.g. w(g_1,g_2) = c_1 g_1 c_2 g_2^{-1}, where the constants c_1,c_2 are elements of G). If such an expression always evaluates to 1 in G, it is called a mixed identity. In recent years, there has been quite some activity in trying to understand mixed identities and their properties in both finite and infinite groups. In this talk, we will focus primarily on infinite groups, discussing mixed identities (or absence thereof) in algebraic groups and properties of the corresponding word maps, and then present an application of the MIF (mixed identity-free) property of groups to questions about maximal discrete subgroups arising from ergodic theory and operator algebras. The talk is based on two pieces of recent joint work: one with Jakob Schneider and the other with Alessandro Carderi, Andreas Thom and Robin Tucker-Drob.
