# Product of trees, quaternions and fake quadrics

### Alina Vdovina (University of Newcastle)

Wednesday 29th January, 2014 16:00-17:00 Maths 204

#### Abstract

We construct an infinite series of simply transitive irreducible lattices in $\PGL_2(\bF_q((t))) \times \PGL_2(\bF_q((t)))$ by means of a quaternion algebra over $\bF_q(t)$. The lattices depend  on an odd prime power $q = p^r$ and a parameter $\tau \in \bF_q^\ast, \tau \not= 1$, and are the fundamental group of a square complex with just one vertex and universal covering $T_{q+1} \times T_{q+1}$, a product of trees with constant valency $q+1$.

Our lattices give rise  to smooth projective surfaces of general type over $\bF_q((t))$ . For $q = 3$, the Zariski-Euler characteristic attains its minimal value $\chi = 1$: the surface is a non-classical fake quadric."