Let us consider the special case when vector $v$ is a lattice vector, i.e. $v\in\Lambda_{2,18}$. In A-model picture it corresponds to the integrality of the class $[\omega]$ of symplectic 2-form. In B-model this means that the non-archimedean field $K$ has valuation in ${\bf Z}\subset{\bf R}$. In terms of ${\bf Z}$-affine structures it means that the monodromy of the affine connection is reduced to $SL(2,{\bf Z})\ltimes{\bf Z}^{2}$. Group $\Gamma_{v}$ is a subgroup (and also a quotient group) of an arithmetic subgroup in the Lie group $SO(1,18)$. Also in this case there is a $\Gamma_{v}$-invariant notion of a point with integer coordinates on $B\simeq S^{2}$, as well of points with coordinates in $\frac{1}{N}{\bf Z}$ for any integer $N\geq 1$. The number $M_{v,N}$ of such points is finite. It is not hard to see that $M_{v,N}=Area_{v}+2={{(v,v)}\over{2}}N^{2}+2$ where $Area_{v}$ is the area of $B$ with a ${\bf Z}$-PL structure corresponding to $v$. This is analogous to the Riemann-Roch formula $rk\,\left(\Gamma(X_{\bf C},L^{\otimes N})\right)=\int_{X}{{c_{1}(L)^{2}}\over{2}}+2$ for an ample line bundle $L$ on a complex K3 surface $X_{\bf C}$.

The action of $\Gamma_{v}$ on $S^{2}$ gives rise to a homomorphism $\Gamma_{v}\to S_{M_{v,n}}$ where $S_{M_{v,n}}$ is the symmetric group. Also the action gives a homomorphism from $\Gamma_{v}$ to the mapping class group $\pi_{1}({\cal{M}}_{0,M_{v,N}}^{unord})$, the fundamental group of the moduli space of genus zero complex curves with $M_{v,N}$ unordered distinct marked points. The last group is closely related to the braid group. The conclusion is that we have constructed homomorphisms from arithmetic groups to a tower of braid groups.

One can deduce from Torelli theorem an interpretation of $\Gamma_{v}$ as a quotient group of the fundamental group of a neighborhood $U$ of a cusp in 19-dimensional moduli space of polarized complex algebraic K3-surfaces, where vector $v$ corresponds to the polarization. Therefore the homomorphism $\Gamma_{v}\to S_{M_{v,n}}$ gives a finite covering $U^{\prime}$ of $U$. One may wonder whether there exists a line bundle over $U^{\prime}$ whose direct image to $U$ coinsides with the direct image of the sheaf $L^{\otimes N}$ from the universal family of K3 surfaces (this question is in spirit of some ideas of Andrey Tyurin, see e.g. [Tyu]).