Simple totally disconnected locally compact groups separated by finiteness properties
Laura Bonn, Sebastian Giersbach
TL;DR
This work extends finiteness-properties separations from discrete to totally disconnected locally compact groups by constructing simple non-discrete tdlc groups that separate $F_n$ and $FP_n$-type properties. The core method uses Smith universal groups $\mathcal{U}(M,N)$ acting on biregular trees with local actions $M$ and $N$, and a finiteness-properties transfer theorem that ties the global finiteness of $\mathcal{U}(M,N)$ to the local finiteness of $M$ and $N$, including a generalization of Haglund–Wise to the tdlc context. The authors produce explicit simple non-discrete tdlc groups of type $F_{n-1}$ but not $F_n$ for each $n$, and a simple group of type $FP_2$ over $\mathbb{Z}$ that is not compactly presented, via Bestvina–Brady groups and carefully chosen automorphism actions. These results yield a versatile framework for constructing simple non-discrete tdlc groups with a wide range of finiteness properties and demonstrate the robustness of finiteness transfers from local to global actions in the tdlc setting.
Abstract
We construct a sequence of simple non-discrete totally disconnected locally compact (tdlc) groups separated by finiteness properties; that is, for every positive integer $n$ there exists a simple non-discrete tdlc group that is of type $F_{n-1}$ but not of type $F_n$. This generalizes a result for discrete groups of Skipper--Witzel--Zaremsky. Furthermore, we construct a simple non-discrete tdlc group that is of type $FP_2$ over $\mathbb{Z}$ but not compactly presented. Our examples arise as Smith universal groups $\mathcal{U}(M, N)$ associated to permutation groups $M$ and $N$. We generalize a theorem of Haglund--Wise to tdlc groups and show that under mild conditions on $M$ and $N$ the finiteness properties of $\mathcal{U}(M, N)$ reflect those of its local actions $M$ and $N$.