All mixed identities are singular in groups with no algebraicity

Abstract

We show that if a group $G$ admits an action with no algebraicity then all of its mixed identities are singular. Previously, such groups were only known to be lawless by a theorem of Abért. 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 $(\mathbb{Q}; <)$. Outside the oligomorphic context, it moreover applies to much-investigated groups, e.g. to Thompson&#39;s groups $F$, $T$, and $V$, to Grigorchuk&#39;s group, and to the homeomorphism groups of any manifold of dimension $\geq 1$. More generally, we prove that all mixed identities of a group $G$ are singular as long as $G$ has an action satisfying certain geometric conditions. This additionally covers the infinite-dimensional general and projective linear groups, recovering e.g. results of Bradford, Schneider, and Thom.