Algebra Seminar talk
2026-10-02
Michael Pinsker
All mixed identities are singular in groups with no algebraicity
Abstract:
I will present a recent joint result with Paolo Marimon. No previous knowledge on the topic beyond the definition of a group is required to follow the talk.
A mixed equation over a group G is an equation of the form w=1, where w is a group word built (using multiplication and inverses) from some variables x, y, z,... and group elements a,b,c,... . A mixed equation is called a mixed identity if universally quantifying its variables yields a true statement in G, in other words, the equation holds whenever we substitute the variables by group elements. "Mixed" refers to the presence of constants (i.e. group elements) in the equation; identities without constants are called laws.
A mixed identity over G is singular if replacing all constants by the identity element one obtains a trivial equation (i.e. the variables cancel out and we obtain the equation 1=1). For example, the singular identity x a x^{-1} a^{-1}=1 expresses that the group element a commutes with all x; the regular identity x x=1 expresses that all group elements have order at most 2.
A permutation group G has no algebraicity if all stabilizers of G of finitely many elements have only infinite orbits (outside the stabilized elements). We show that if a group admits an action as a permutation group with no algebraicity then all of its mixed identities are singular. Our result confirms, in particular, a conjecture of Bodirsky, Schneider, and Thom for a large class of oligomorphic permutation groups. It thereby not only subsumes numerous results from the literature in a simple uniform theorem, but also settles the question for prominent groups for which the conjecture was an open problem, such as the automorphism group of (ℚ;<). The result also applies outside the oligomorphic context, e.g. to Thompson's groups F,T, and V, to Grigorchuk's group, and to the homeomorphism groups of any manifold of dimension ≥1. More generally, we prove that all mixed identities of a group are singular as long as it has an action satisfying certain geometric conditions. This additionally covers, for example, the infinite-dimensional general and projective linear groups.