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

### Inna Capdeboscq, Dmitriy Rumynin and Anne Thomas

#### Abstract

Let $$K$$ be the field of formal Laurent series over the finite field of order $$q$$. We construct cocompact lattices $$\Gamma'_0 < \Gamma_0$$ in the group $$G = \mathrm{PGL}_d(K)$$ which are type-preserving and act transitively on the set of vertices of each type in the building associated to $$G$$. The stabiliser of each vertex in $$\Gamma'_0$$ is a Singer cycle and the stabiliser of each vertex in $$\Gamma_0$$ is isomorphic to the normaliser of a Singer cycle in $$\mathrm{PGL}_d(q)$$. We then show that the intersections of $$\Gamma'_0$$ and $$\Gamma_0$$ with $$\mathrm{PSL}_d(K)$$ are lattices in $$\mathrm{PSL}_d(K)$$, and identify the pairs $$(d,q)$$ such that the entire lattice $$\Gamma'_0$$ or $$\Gamma_0$$ is contained in $$\mathrm{PSL}_d(K)$$. Finally we discuss minimality of covolumes of cocompact lattices in $$\mathrm{SL}_3(K)$$. Our proofs combine a construction of Cartwright and Steger with results about Singer cycles and their normalisers, and geometric arguments.

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 Tuesday, June 26, 2012