arXiv subjects

Sebastian Giersbach

Publications and source records attributed to Sebastian Giersbach.

2 recordsLinked to original sources

arXiv2026

Quasi-isometries between graphs of totally disconnected locally compact groups

Let $G$ and $H$ be compactly generated totally disconnected locally compact (tdlc) groups that decompose as finite graphs of tdlc groups $(\mathcal{G}, \mathcal{A})$ and $(\mathcal{H}, \mathcal{B})$ such that all edge groups are compact and all vertex groups have at most one end. We generalize a result of Papasoglu--Whyte to tdlc groups and show that $G$ and $H$ are quasi-isometric if and only if they have the same number of ends and every one-ended vertex group of $(\mathcal{G}, \mathcal{A})$ is quasi-isometric to a one-ended vertex group of $(\mathcal{H}, \mathcal{B})$, and vice versa. As an application, we construct uncountably many pairwise non-quasi-isometric compactly generated non-discrete simple tdlc groups, strengthening a result by Smith.

math.GR↗

arXiv2025

Simple totally disconnected locally compact groups separated by finiteness properties

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 $N$.

math.GR↗