the building. However, no presentations of these lattices are given.

Again, we do not know whether the lattices we construct here are commensurable to the examples given by Kantor.

### Buildings of type \~A2

For buildings of type $\tilde A_2$, we obtain the following result. Fix three Singer projective planes (Singer generalised triangles) of order $q$ along with three *Singer groups* $S_i$, $i\in\{1,2,3\}$, that is, groups acting regularly on points (and hence on lines) of these planes. For each of these planes, fix a point $p_i$ and a line $l_i$ and write $$D_i = \{ d\in S_i: p_i\text{ is incident to } d(l_i) \}.$$ These sets are called *difference sets*. Finally, write $J=\{0,1,\ldots,q\}$ and fix three bijections $d_i:J\rightarrow D_i$.

There is an $\tilde A_2$-building $X$ such that the group $$\Gamma_1 = \Bigl\langle S_1,S_2,S_3 \,\Big| \begin{array}{c}\text{ all relations in the groups }S_1,S_2,S_3,\\ d_1(j)d_2(j)d_3(j) = d_1(j')d_2(j')d_3(j') \quad\forall j,j'\in J\end{array}\Bigr\rangle$$ is a uniform lattice in the full automorphism group of the building. The union of all chambers containing a fixed panel is a fundamental domain for the action.

Conversely, every panel-regular lattice on a building of type $\tilde A_2$ arises in this way and admits a presentation of the above form with $d_1(0)d_2(0)d_3(0)=1$.

