Generalisable presentations and compactness properties of locally compact right-angled Artin groups

Abstract

We propose the systematic study of presentations that can be generalised over a continuous open group monomorphism. Presentations with this property can turn well-known presentations such as those for as orientable surface groups, Artin groups, and some Thompson groups, into topological groups with a prescribed open subgroup. Later we focus on right-angled Artin groups (RAAGs) and introduce a notion of topological RAAGs. Our approach differs from lattice envelopes and produces examples of locally compact (LC) groups that contain RAAGs as discrete subgroups, but generally not as lattices. We investigate some geometric aspects of topological RAAGs, with a special emphasis on compactness properties of LC ones. This includes a study of universal Salvetti-type complexes which may be of independent interest. These complexes share some properties with buildings. Although in some cases they are CAT(0) cube complexes and provide models for classifying spaces, in other cases they are not even uniquely geodesic. For a large class of examples we establish high connectivity properties for these complexes. This yields novel examples of LC groups with prescribed compactness properties or rational cohomological dimension. We note that the Bestvina-Brady machinery does not automatically generalise to this setting; nevertheless, we extend the Bieri-Stallings construction to obtain totally disconnected locally compact (TDLC) groups of type $FP_n$ but not $FP_{n+1}$. Along the way we record counterparts of cohomological results, such as a Mayer-Vietoris sequence and Künneth formula in discrete (co)homology for TDLC groups, which have not appeared elsewhere in the literature. Despite our non-discrete LC focus we obtain, as by-product, new examples of discrete groups with controlled finiteness properties including, for every $n \geq 1$, a Thompson-like Bieri-Stallings group of type $F_n$ but not $F_{n+1}$.