# 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.