The Tits alternative for visibility spaces

TL;DR

The paper addresses whether finitely generated groups acting properly discontinuously by isometries on visibility CAT($0$) spaces with bounded packing satisfy the Tits alternative. It shows that such groups are either almost nilpotent or contain a nonabelian free subgroup of rank $2$, with the almost nilpotent case equivalent to having $|\mathcal{L}(\Gamma)|\le 2$ in the boundary $X(\infty)$. The proof uses a ping-pong argument when the limit set has at least three points, and an iterated transverse-space approach combined with a generalized Margulis lemma (Breuillard, Green, Tao) to deduce almost nilpotency; torsion cases yield finite groups with a global fixed point. This generalizes Ji–Wu results, resolves torsion questions for visibility spaces under bounded packing, and extends Tits-type dichotomies to a broad nonpositively curved setting with implications for boundary dynamics and group actions.

Abstract

Let $Γ$ be a finitely generated group acting properly discontinuously by isometries on a visibility CAT(0) space $X$ that satisfies the bounded packing property. We prove that $Γ$ satisfies the Tits alternative: it is either almost nilpotent or contains a free nonabelian subgroup of rank $2$. In the former case, it is equivalent to that the cardinality of the limit set of $Γ$ in the geometric boundary of $X$ is no greater than $2$. As an application of the Tits alternative, we show that any finitely generated torsion group acting properly discontinuously by isometries on such a space must be a finite group and have a global fixed point.

Theorems & Definitions (10)

• Theorem 1.2
• Remark 1.3
• Theorem 1.5
• Lemma 2.1
• proof
• Lemma 2.2
• Definition 2.3
• Lemma 2.4
• Lemma 2.5
• Definition 2.6