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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Arithmetic groups of higher $ {\bf Q}$-rank cannot act on $ 1$-manifolds


Author: Dave Witte
Journal: Proc. Amer. Math. Soc. 122 (1994), 333-340
MSC: Primary 22E40; Secondary 20F60, 20H05, 57S25
MathSciNet review: 1198459
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Abstract: Let $ \Gamma $ be a subgroup of finite index in $ {\text{SL}_n}(\mathbb{Z})$ with $ n \geq 3$. We show that every continuous action of $ \Gamma $ on the circle $ {S^1}$ or on the real line $ \mathbb{R}$ factors through an action of a finite quotient of $ \Gamma $. This follows from the algebraic fact that central extensions of $ \Gamma $ are not right orderable. (In particular, $ \Gamma $ is not right orderable.) More generally, the same results hold if $ \Gamma $ is any arithmetic subgroup of any simple algebraic group G over $ \mathbb{Q}$, with $ \mathbb{Q}$   -$ {\text{rank}}(G) \geq 2$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1994-1198459-5
PII: S 0002-9939(1994)1198459-5
Article copyright: © Copyright 1994 American Mathematical Society