Journal of Lie Theory 24 (2014), No. 4, 957--967
Copyright Heldermann Verlag 2014
Limits of Contraction Groups and the Tits Core
Pierre-Emmanuel Caprace
Université Catholique de Louvain, Chemin du Cyclotron 2, Bte L7.01.02, 1348 Louvain-la-Neuve, Belgium
pe.caprace@uclouvain.be
Colin D. Reid
School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, NSW 2308, Australia
Colin.Reid@newcastle.edu.au
George A. Willis
School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, NSW 2308, Australia
George.Willis@newcastle.edu.au
The Tits core G+ of a totally disconnected locally compact group G is defined as the abstract subgroup generated by the closures of the contraction groups of all its elements. We show that a dense subgroup is normalised by the Tits core if and only if it contains it. It follows that every dense subnormal subgroup contains the Tits core. In particular, if G is topologically simple, then the Tits core is abstractly simple, and when G+ is non-trivial, it is the smallest dense normal subgroup. The proofs are based on the fact, of independent interest, that the map which associates to an element the closure of its contraction group is continuous.
Keywords: Totally disconnected locally compact group, simple group, contraction group, Chabauty topology.
MSC: 22D05, 20E32
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