Paolo Marimon (University of Oxford) – Mixed identities and Neumann's lemma (LYMoTS)

This is joint work with Michael Pinsker. A mixed identity for a group $G$ is a word $w(x_1, \dots, x_r, g_1,\dots, g_n)$ in the language of groups (with variables $x_1,\dots, x_r$ and constants $g_1, \dots, g_n\in G$) such that for any $h_1, \dots, h_r\in G$, $w(h_1, \dots, h_r, g_1,\dots, g_n)=1$. For example, in an Abelian group, $x y x^{-1} y^{-1}$ is a mixed identity (without constants). A mixed identity is singular if forgetting the constants and reducing the resulting resulting word, we get the identity. For example, $x g x^{-1}$ is singular, but $x g x$ is not. Recently, Bodirsky, Schneider, and Thom conjectured that if $G$ is the automorphism group of an $\omega$-categorical structure, then all of its mixed identities are singular. We prove that if $G$ has an action with no algebraicity, then all of its mixed identities are singular. Our result applies to the automorphism groups of a large class of $\omega$-categorical structures, including $(\mathbb{Q}, <)$, for which the aforementioned conjecture was open, but also to several other groups of interest to geometric group theory, such the Thompson group $F$, whose mixed identities were studied in works of Ivanov, Słanina, and Zarzycki, or the homeomorphism groups of manifolds. More generally, we prove that all mixed identities of a group $G$ are singular as long as $G$ admits an action for which algebraic closure forms a modular pregeometry and satisfies a certain higher dimensional variant of Neumann's lemma. This covers also infinite vector spaces over finite fields, whose mixed identities were studied by Bradford, Schneider, and Thom by different methods.

This talk is part of the Lancashire Yorkshire Model Theory Seminar.