Duke Mathematical Journal

Trees, contraction groups, and Moufang sets

Pierre-Emmanuel Caprace and Tom De Medts

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We study closed subgroups G of the automorphism group of a locally finite tree T acting doubly transitively on the boundary. We show that if the stabilizer of some end is metabelian, then there is a local field k such that PSL2(k)GPGL2(k). We also show that the contraction group of some hyperbolic element is closed and torsion-free if and only if G is (virtually) a rank 1 simple p-adic analytic group for some prime p. A key point is that if some contraction group is closed, then G is boundary-Moufang, meaning that the boundary T is a Moufang set. We collect basic results on Moufang sets arising at infinity of locally finite trees, and we provide a complete classification in case the root groups are torsion-free.

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Duke Math. J., Volume 162, Number 13 (2013), 2413-2449.

First available in Project Euclid: 8 October 2013

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Zentralblatt MATH identifier

Primary: 20E08: Groups acting on trees [See also 20F65]
Secondary: 20E42: Groups with a $BN$-pair; buildings [See also 51E24] 20G25: Linear algebraic groups over local fields and their integers 22D05: General properties and structure of locally compact groups


Caprace, Pierre-Emmanuel; De Medts, Tom. Trees, contraction groups, and Moufang sets. Duke Math. J. 162 (2013), no. 13, 2413--2449. doi:10.1215/00127094-2371640. https://projecteuclid.org/euclid.dmj/1381238849

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