**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

### Thomas

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

###### Speaker:

Anne Thomas (University of Sydney)

###### Title:

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

###### Abstract:

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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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 John.Enyang@sydney.edu.au

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