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This is a Quantitative Study study.

October 3, 2025Open Access

Simple totally disconnected locally compact groups separated by finiteness properties

Puntos clave

This research constructs simple non-discrete tdlc groups that exhibit distinct finiteness properties.
A sequence of simple groups is identified where each group is of type F_n-1 but not type FP_n.
The examples arise as Smith groups associated with pairs of permutation groups M and N.
A generalization of a prior theorem is provided, showing finiteness properties reflect those of local groups.

Resumen

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₍-₁ but not of type FPₙ. This generalizes a result of Skipper--Witzel--Zaremsky for discrete groups. Furthermore, we construct a simple non-discrete tdlc group that is of type FP₂ but not compactly presented. Our examples arise as Smith groups U (M, N) associated to pairs of permutation groups M and N. We generalize a theorem of Haglund--Wise for a special case and show that under mild conditions the finiteness properties of U (M, N) reflect those of its local groups M and N, and vice versa.

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Cite This Study

Bonn et al. (Fri,) studied this question.

synapsesocial.com/papers/68e02f46f0e39f13e7fa2e74 https://doi.org/https://doi.org/10.48550/arxiv.2509.05101