Isotropic nonarchimedean S-arithmetic groups are not left orderable

2004/05/28 by Lucy Lifschitz, Dave Witte Morris, Dave Morris +2
Mathematics · #06F15 #11F06 #20F60 #20F65 #22E40 #57S25 #Advanced Topics in Algebra #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #advanced mathematical theories #math.GR #msc:06F15 #msc:11F06 #msc:20F60 #msc:20F65 #msc:22E40 #msc:57S25

paper · pdf · doi:10.48550/arxiv.math/0405536

4 pages, no figures; new title, abstract and introduction to reflect improved conclusions in the nonarchimedean case

openalex publication_date 2004/05/28 · arxiv created 2004/06/03 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let A = Z[c], where c is an irrational number whose square is rational, or let A = Z[1/r], where r > 1 is a square-free natural number. We show that no finite-index subgroup of SL(2,A) is left orderable. (Equivalently, these subgroups have no nontrivial orientation-preserving actions on the real line.) This implies that if G is an isotropic F-simple algebraic group over an algebraic number field F, then no nonarchimedean S-arithmetic subgroup of G is left orderable. Our proofs are based on the fact, proved by B.Liehl, that every element of SL(2,A) is a product of a bounded number of elementary matrices.

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