Tits Alternative for groups acting properly on 2-dimensional recurrent complexes

Tits Alternative for -dimensional complexes

“…Regarding Corollary C, for right-angled Artin groups, the Tits Alternative follows from the work of Baudisch [Bau81]. In our previous work [OP21], we showed the Tits Alternative for a subclass of 2-dimensional Artin groups, containing all large-type Artin groups. Recently, in [MP22], we proved the Tits Alternative for 2-dimensional Artin groups of hyperbolic type, and in [MP20], we proved it for FC-type Artin groups.…”

Tits Alternative for 2-dimensional CAT(0) complexes

“…Regarding Corollary C, for right-angled Artin groups the Tits Alternative follows from the work of Sageev and Wise [SW05]. In our previous work [OP21] we showed the Tits Alternative for a subclass of 2-dimensional Artin groups, containing all large-type Artin groups. Recently, in [MP21b] we proved the Tits Alternative for 2-dimensional Artin groups of hyperbolic type, and in [MP20] we proved it for FC-type Artin groups.…”

“…This property might be seen as the first step towards the Tits Alternative. In [OP21] we proved the Tits Alternative for the class of 2-dimensional recurrent complexes. This class contains all 2-dimensional Euclidean buildings, 2-dimensional systolic complexes, as well as some complexes outside the CAT(0) setting.…”

“…Organisation. In Section 2 we prove Theorem E. In Section 3 we recall the method of invariant cocompact subcomplexes from [OP21], which allows us to reduce Theorem A to Proposition 3.2 that assumes the existence of edges of degree ≥ 3 in our complex X. Under this assumption we can exclude the cases of virtually cyclic or Z 2 groups in Section 4.…”

Fixed points for group actions on 2-dimensional affine buildings

“…Over the years Tits Alternatives have been proved for many other classes of groups, for example groups acting on CAT(0) cubical complexes [23] or on median rank spaces [7]. Osajda and Przytycki [17] built on Theorem 1.3 to prove a Tits Alternative for groups with a bound on the order of finite subgroups which act properly on 2-dimensional recurrent complexes.…”