4 problems

Rosenberger's conjecture. If $G$ is a generalised triangle group, then either $G$ is virtually solvable or $G$ contains a non-abelian free subgroup.

A group is understood here in the usual sense, and the Tits Alternative is the dichotomy asserting that a group, or its finitely generated subgroups, either contains a non-abelian…

Let $X$ be a CAT(0) metric space, and let $G$ act on $X$ properly and cocompactly. The CAT(0) Tits alternative conjecture. The group $G$ satisfies the strong Tits alternative: ever…

Let $M$ be a compact manifold and let $\Gamma$ be a finitely generated group of smooth diffeomorphisms of $M$. Ghys's alternative conjecture. Either $\Gamma$ contains a free group…