SMS scnews item created by John Enyang at Tue 22 May 2012 1149
Type: Seminar
Distribution: World
Expiry: 26 May 2012
Calendar1: 25 May 2012 1205-1255
CalLoc1: Carslaw 175
Auth: enyang@penyang.pc (assumed)

Algebra Seminar

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


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


Anne Thomas (University of Sydney)


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.


After the seminar we will take the speaker to lunch.

See the Algebra Seminar web page for information about other seminars in the series.

John Enyang

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