Journal of Lie Theory 30 (2020), No. 4, 965--980
Copyright Heldermann Verlag 2020
Topologically Simple, Totally Disconnected, Locally Compact Infinite Matrix Groups
Peter Groenhout
The University of Newcastle, Callaghan 2308, NSW, Australia
peter.groenhout@uon.edu.au
George A. Willis
The University of Newcastle, Callaghan 2308, NSW, Australia
george.willis@newcastle.edu.au
Colin D. Reid
The University of Newcastle, Callaghan 2308, NSW, Australia
colin@reidit.net
We construct uncountably many non-locally isomorphic examples of topologically simple nondiscrete totally disconnected locally compact groups. The new examples differ from known examples of such groups in that they have trivial quasi-centre, but also have infinite abelian locally normal subgroups. The examples are constructed as almost upper-triangular matrices modulo scalar matrices over finite fields, where "almost upper-triangular" is defined with respect to one of an uncountable family of preorders generalising the natural orders on the set of integers and the set of natural numbers.
Keywords: Infinite matrix, finite field, locally compact group, topologically simple, quasi-centre.
MSC: 22D05; 20H30, 20E18.
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