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Isotropic nonarchimedean S-arithmetic groups are not left orderable

Lucy Lifschitz·Dave Morris
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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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Lucy Lifschitz, Dave Morris. 2004-06-03. Isotropic nonarchimedean S-arithmetic groups are not left orderable. https://arxiv.org/abs/math/0405536

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