We construct an uncountable sequence of groups acting uniformly properly on hyperbolic spaces. show that only countably many these can be virtually torsion-free. This gives new examples spaces are not torsion-free and cannot subgroups groups.

Sets with a self-distributive operation (in the sense of ( ◃ b ) c = ), in particular quandles, appear knot and braid theories, Hopf algebra classification, study Yang–Baxter equation, other areas. An important invariant quandles is their structure group. The group finite quandle known to be either “boring” (free abelian), or “interesting” (non-abelian torsion). In this paper we explicitly desc...

We show that there exist non-unitarizable groups without non-abelian free subgroups. Both torsion and torsion free examples are constructed.

A monoid M is a Cale monoid with base Q if for every nonunit x ∈ M there exists a positive integer n such that xn factors uniquely up to order and associates as elements from Q ⊆ M\M. An integral domain D is a Cale domain with base Q, if its multiplicative monoid of nonzero elements is a Cale monoid with base Q. We explore the basic properties of Cale monoids and integral domains. In particular...