Simple totally disconnected locally compact groups separated by finiteness properties

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 $N$.