Combinatorial description of the principal congruence subgroup $\gamma$(2) in SL(2, z)

TL;DR

This paper provides a combinatorial characterization of certain sequences of integers that correspond to matrices in the principal congruence subgroup of level 2 in SL(2, Z), using polygon dissections.

Contribution

It introduces a novel combinatorial description linking integer sequences to polygon dissections for matrices in the principal congruence subgroup.

Findings

Characterization of sequences via polygon dissections

Connection between matrix properties and geometric dissections

Explicit criteria for membership in the principal congruence subgroup

Abstract

We characterize sequences of positive integers (c 1 , c 2 , ..., cn) for which the (2 x 2)-matrix c 1 --1 1 0 $\times$ $\times$ $\times$ cn --1 1 0 belongs to the principal congruence subgroup of level 2 in SL(2, Z). The answer is given in terms of dissections of a convex n-gon into a mixture of triangles and quadrilaterals.