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

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 $\mathrm{\Gamma }\le G$ such that $G/\mathrm{\Gamma }$ 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 $\mathrm{\Gamma }$ of $G$ is a cocompact lattice exactly when it acts cocompactly on $X$ with finite stabilisers. We construct a cocompact lattice ${\mathrm{\Gamma }}_0$ 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 ${\mathrm{\Gamma }}_0$ 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