Suborbital Graphs for a Special Subgroup of the $SL(3,\mathbb Z)$

Murat Beşenk

In this paper we examine some properties of suborbital graphs for the group $SL^*(3,\mathbb Z)$. We first introduce an invariant equivalence relation by using the congruence subgroup $SL^*(3,\mathbb Z)$ instead of $\Gamma_0(n)$ and obtain some results for the newly constructed subgraphs $F_{u,n}$ whose vertices form the block $[\infty]$. We obtain edge and circuit conditions and some relations between lengths of circuits in $F_{u,n}$ and elliptic elements of $\Gamma_0(n)$.