Non-sofic Tarski monsters

Abstract

We prove the existence of non-sofic Tarski monsters of every sufficiently large prime exponent and of torsion-free non-sofic Tarski monsters. The proof combines a theorem of Sandhu–Thom with quotient theorems of Semenov. These groups are minimal non-sofic, since all their proper subgroups are cyclic. A theorem of Minasyan–Osin–Witzel gives continuum many pairwise non-quasi-isometric torsion-free simple examples. Kulikova's quotient theorem also yields non-simple minimal non-sofic groups of exponent $p^m$ for every $m\geq2$ and every sufficiently large prime $p$. We then study descending chains of non-sofic subgroups in other constructions. We give proper self-embeddings of the binary Leavitt unit group, higher Brin–Thompson groups and the hyperbolic group studied by Douglas and Mghirbi. The explicit Kun–Thom wreath products and doubles are locally graded. These examples are $ω$-non-sofic, and the locally graded ones contain no minimal non-sofic subgroup.