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)

# Cocompact lattices on $$\tilde{A}_n$$ buildings

### Thomas

###### 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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