A special case arises as follows: Classical projective planes admit cyclic Singer groups. Denote the generators of the three Singer groups $S_i$ by $\sigma_i$, and write $$\Delta_i=\{ \delta \in {\mathbb Z}/ |S_i| : \sigma_i^\delta \in D_i\}.$$ These are difference sets in the classical sense.

For any prime power $q$, for any three classical difference sets $\Delta_1$, $\Delta_2$, $\Delta_3$ containing 0, and for any bijections $\delta_i:J\rightarrow \Delta_i$ satisfying $\delta_i(0)=0$, the group $\Gamma_2$ with presentation $$\Gamma_2 = \langle \sigma_1, \sigma_2, \sigma_3 \,|\, \sigma_1^{q^2+q+1}=\sigma_2^{q^2+q+1}=\sigma_3^{q^2+q+1}=1,\, \sigma_1^{\delta_1(j)}\sigma_2^{\delta_2(j)}\sigma_3^{\delta_3(j)} = 1 \quad\forall j \in J\rangle$$ is a uniform lattice in a building of type $\tilde A_2$.

Two examples are: $$\begin{aligned}
    \Lambda &= \langle \sigma_1,\sigma_2,\sigma_3 \,|\, \sigma_1^7=\sigma_2^7=\sigma_3^7 = \sigma_1\sigma_2\sigma_3 = \sigma_1^3\sigma_2^3\sigma_3^3 = 1\rangle,\\
    \Lambda' &= \langle \sigma_1,\sigma_2,\sigma_3 \,|\, \sigma_1^{13}=\sigma_2^{13}=\sigma_3^{13} = \sigma_1\sigma_2^3\sigma_3^9 = \sigma_1^3\sigma_2^9\sigma_3 =\sigma_1^9\sigma_2\sigma_3^3 = 1\rangle.\end{aligned}$$ Further examples can easily be constructed using the list of difference sets found in the La Jolla Difference Set Repository [@LaJolla].

If we start from cyclic Singer groups, we obtain very explicit descriptions of the associated buildings. Using these, we can give explicit descriptions of the spheres of radius two in these buildings, which lead to an interesting incidence structure called a *Hjelmslev plane of level two*. Using a result of Cartwright-Mantero-Steger-Zappa 