Finiteness properties of totally disconnected locally compact groups

Abstract

In this paper we investigate finiteness properties of totally disconnected locally compact groups for general commutative rings R, in particular for R = Z and R = Q. We show these properties satisfy many analogous results to the case of discrete groups, and we provide analogues of the famous Bieri's and Brown's criteria for finiteness properties and deduce that both FPn-properties and Fn-properties are quasi-isometric invariant. Moreover, we introduce graph-wreath products in the category of totally disconnected locally compact groups and discuss their finiteness properties.Proposition 4.5. Suppose that a t.d.l.c. group G admits an n-good G-CWcomplex X over R of type F n , i.e., such that the n-skeleton has finitely many G-orbits. Then G is of type FP n .Proof. The proof of [11, Proposition 1.1] can be transferred verbatim.Remark 4.6. Notice that [13, Proposition 6.6] can be deduced from this result considering the topological realisation of a discrete simplicial G-complex.

A Stallings-Swan theorem for unimodular totally disconnected locally compact groups

“…a proper discrete G-CW-complex X such that X is contractible and, for all O ∈ CO(G), the fixed-point set X O is non-empty and contractible. According to [5], a t.d.l.c. group G is of type F n (with 0 ≤ n < +∞) if there exists a contractible proper discrete G-CW-complex X such that G acts on the nskeleton of X with finitely many orbits.…”

“…group G satisfying cd Q (G) ≤ 1 and an extra finiteness condition, such as compact presentability (which is equivalent to being of type F 2 (cf. [5,Proposition 3.4])) or of being CO-bounded, implies that G is accessible in the sense above. At first sight, the just-mentioned two finiteness conditions seem unrelated.…”

Computable topological abelian groups

“…This means that, in particular, the a i,j which were truly independent remain independent when we redefine them in (c). (This follows from an elementary analysis of their respective linear spans; we leave the elementary details to the reader 4 .) The minimality of indices also implies that, in the limit, every element of A 0 will be in the linear span of D 0 .…”

“…The class of totally disconnected locally compact (t.d.l.c.) groups is perhaps the narrowest class extensively studied in the literature (recent papers include [58,57,20,4,29]) which contains both countable discrete and profinite Polish groups. In the companion paper [37] we suggest three different reasonable definitions of computability for a t.d.l.c.…”