A team of mathematicians has proved that groups called Tarski monsters, which are normally built inside the theory of sofic groups, can be constructed that are not sofic. The proof shows that for every sufficiently large prime exponent p there is a non‑sofic Tarski monster of exponent p, and a separate construction gives a torsion‑free non‑sofic Tarski monster.

The construction starts with a non‑trivial torsion‑free hyperbolic group H that has no non‑trivial sofic quotient, a result of Sandhu and Thom. Semenov’s quotient theorems then produce Tarski monsters as quotients of this H. Because the quotient inherits the lack of a sofic quotient, the resulting group is non‑sofic.

All proper subgroups of these monsters are cyclic, so they are minimal non‑sofic: every non‑trivial subgroup is sofic. In the torsion‑free case the authors generate a continuum (2ℵ0) of pairwise non‑quasi‑isometric simple examples, all of them minimal non‑sofic.

These results give the first explicit families of non‑sofic groups that are minimal in the sense of subgroup structure and also display richness in their quasi‑isometry types.