2022/04/04 by Tobias Hartnick, Hartnick, Tobias, Stefan Witzel +1
Computer Science · Mathematics · #20G30 #20G35 #51E24 #52C23 #57M07 #Advanced Topology and Set Theory #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR) #Metric Geometry (math.MG) #Number Theory (math.NT) #Primary 20F65 #Secondary 11F75 #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2204.01496
openalex publication_date 2022/04/04 · openalex created_date 2022/05/05 · openalex updated_date 2026/07/28
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, Köhl 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.