A new lattice invariant for lattices in totally disconnected locally compact groups
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A NEW LATTICE INVARIANT FOR LATTICES IN TOTALLY DISCONNECTED LOCALLY COMPACT GROUPS BY
Bruno Duchesne∗ ´ Cartan de Lorraine Institut Elie CNRS and Universit´e de Lorraine, F-54052 Nancy, France e-mail: [email protected] AND
Robin Tucker-Drob∗∗ Department of Mathematocs, Texas A&M University College Station, TX 77843-3368, USA e-mail: [email protected] AND
Phillip Wesolek Department of Mathematics and Computer Science, Wesleyan University, Middletown, CT 06459, USA e-mail: [email protected]
ABSTRACT
We introduce and explore a natural rank for totally disconnected locally compact groups called the bounded conjugacy rank. This rank is shown to be a lattice invariant for lattices in sigma compact totally disconnected locally compact groups; that is to say, for a given sigma compact totally disconnected locally compact group, some lattice has bounded conjugacy rank n if and only if every lattice has bounded conjugacy rank n. Several examples are then presented.
∗ B.D. is supported in part by French projects ANR-14-CE25-0004 GAMME and
ANR-16-CE40-0022-01 AGIRA.
∗∗ R.T.D. is supported in part by NSF grants DMS 1600904 and DMS 1855825.
Received November 28, 2018 and in revised form September 27, 2019
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B. DUCHESNE, R. TUCKER-DROB AND P. WESOLEK
Isr. J. Math.
1. Introduction A property P of groups is called a lattice invariant if whenever Δ and Γ are lattices in a locally compact group G, then Δ has P if and only if Γ has P . Lattice invariants are closely related to invariants of measure equivalence, where two discrete groups Γ and Δ are called measure equivalent if they admit commuting actions on a non-zero sigma-finite standard measure space with each action admitting a finite measure fundamental domain; see [10]. Indeed, two lattices in the same locally compact group are measure equivalent, so any invariant of measure equivalence is also a lattice invariant. Amenability and Kazhdan’s property (T) are two fundamental examples of invariants of measure equivalence and so lattice invariants. However, natural examples of lattice invariants appear to be rather rare and examples which are not additionally measure equivalence invariants are all the more rare. In the work at hand, we discover and explore a new lattice invariant for lattices in totally disconnected locally compact (t.d.l.c.) groups, which we call the bounded conjugacy rank. As will become clear, bounded conjugacy rank additionally fails to be an invariant of measure equivalence. This rank appears to be a natural and robust numerical invariant which warrants further exploration. Remark 1.1: For our main theorems, we restrict our attention to sigma compact locally compact groups. Sigma compact locally compact groups cover most locally compact groups of interest. In particular, every compactly generated locally compact group is sigma compact.
1.A. Statement of results. For G a topological group and H a subgroup, the BC-centralizer of H in G is BCG (H) := {g ∈ G | g H is relatively compact}, where g H denotes the set of H conjugat
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