University of Sydney Algebra Seminar

Anne Thomas (University of Sydney)

Friday 25th May, 12:05-12:55pm, Carslaw 175

Cocompact lattices on \(\tilde{A}_n\) buildings

A cocompact lattice in a locally compact group \( G \) is a discrete subgroup \( \Gamma \leq G \) such that \( G / \Gamma \) is compact. Let \( X \) be the building for \( G = \mathrm{PGL}_d(K) \), where \( K \) is the field of formal Laurent series over the finite field of order \(q\). Then a subgroup \( \Gamma \) of \( G \) is a cocompact lattice exactly when it acts cocompactly on \( X \) with finite stabilisers. We construct a cocompact lattice \( \Gamma_0 \) in \(G\) which acts transitively on the set of vertices of each type in \(X\), so that each vertex stabiliser is the normaliser of a Singer cycle in the finite group \( \mathrm{PGL}_d(q)\). We also show that the intersection of \( \Gamma_0 \) with \( H = \mathrm{PSL}_d(K) \) is a cocompact lattice in \(H\), and provide a geometric description of this intersection for certain pairs \( (d,q) \). Our proof uses a construction by Cartwright, Steger, Mantero and Zappa (in the case \( d = 3 \) ) and Cartwright-Steger (for \( d > 3 \) ) of lattices acting simply-transitively on the vertex set of \(X\), which employed cyclic simple algebras. We also use classical results on the action of subgroups of \( \mathrm{PGL}_d(q) \) on the links of vertices in \(X\), which are finite projective geometries. This is joint work with Inna Capdeboscq and Dmitry Rumynin.

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