A Tarski monster is an infinite finitely generated noncyclic group whose proper nontrivial subgroups are all cyclic.
In the prime-exponent form, every proper nontrivial subgroup
has the same prime order 
. Such groups
exist for sufficiently large primes 
(Ol'shanskii 1991). In a torsion-free Tarski monster every
proper nontrivial subgroup is infinite cyclic.
Ersoy (2026, Theorems A and B) reports nonsofic Tarski monsters of every sufficiently large prime exponent, and torsion-free nonsofic examples. The torsion-free examples include continuum many pairwise non-quasi-isometric simple groups. The argument combines quotient theorems with the Sandhu-Thom construction of a hyperbolic group having no nontrivial sofic group quotient. Independent verification of that complete construction has not been established here, so the inherited qualification remains.
