For a permutation group G on a set X, a mixed identity is an expression of the form w(x_1,...,x_n,g_1,...,g_m)=1, where w is a word in the language of groups with constants g_1,...,g_m from G and free variables x_1,...,x_n. We say that the mixed identity holds in G if substitution of arbitrary elements of G for the free variables yields a true statement. The mixed identity is singular if deletion of all constants yields a true statement in the free group: for example, xgx^{-1}=1 (where x is a variable and g a constant) is singular. We consider the question which non-singular mixed identities hold in a given group G. The group G is oligomorphic if it acts with finitely many orbits on X^n, for all n. Based on several examples, Bodirsky, Schneider, and Thom have conjectured that in oligomorphic groups only singular identities can hold. We confirm this conjecture for all oligomorphic G without algebraicity, i.e. with the property that for any tuple t of elements of X, the stabilizer of t in G does not stabilize any element outside t. In fact, we obtain this result under even more general conditions satisfied e.g. by the general linear group of any infinite vector space over a finite field. This is joint work with Paolo Marimon (Oxford).
Mixed identities in oligomorphic groups Sponsored by the Meyer Fund
Mixed identities in oligomorphic groups