SUBGROUPS OF FINITE INDEX IN (2, 3, n)-TRIANGLE GROUPS

Abstract

AbstractFor an integer n ≥ 7, let Δ(n) denote the (2, 3, n)-triangle group, and let M(n) be the positive integer determined by the conditions that Δ(n) has a subgroup of index m for all m ≥ M(n), but no subgroup of index M(n) − 1. The main purpose of the paper is to obtain information (bounds, in some cases explicit values) concerning the function M(n) (cf. Theorem 1). We also show that Δ(n) is replete (i.e., has a subgroup of index m for every integer m ≥ 1) if, and only if, n is divisible by 20 or by 30 (see Theorem 2).