Higher finiteness properties of arithmetic approximate lattices: The Rank Theorem for number fields

TL;DR

This paper extends the Rank Theorem to S-arithmetic approximate groups over number fields, establishing their finiteness properties and showing they depend on the sum of local ranks, with a characteristic-free approach.

Contribution

It introduces geometric and homological finiteness properties for approximate groups and generalizes the Rank Theorem from positive characteristic to characteristic zero.

Findings

Finiteness length is finite for S-arithmetic approximate groups without infinite places.

Explicit computation of the finiteness length as one less than the sum of local ranks.

The proof is characteristic free, highlighting the role of infinite places in finiteness properties.

Abstract

We introduce geometric and homological finiteness properties for countable approximate groups via coarse geometry and then study these finiteness properties for S-arithmetic reductive approximate groups. For S-arithmetic approximate groups without infinite places we show that the finiteness length is finite and compute this finiteness length explicitly. In the simple case it is one less than the sum of the local ranks. This extends the Rank Theorem of Bux, K\"ohl and the second author from positive characteristic to characteristic zero. Our proof is based on a geometric version of their proof, but except for some input from reduction theory it is characteristic free. This indicates that the apparent differences between arithmetic groups in characteristic zero and positive characteristic concerning finiteness properties are entirely due to the presence of infinite places.