The extended hecke groups as semi-direct products and related results

Abstract

The extended Hecke groups H̄(λq) have been worked in (Sahin and Bizim, 2003) as amalgamated free products. In this paper, we first show that H(λq) is the semi-direct product (split extension) of the Hecke group H̄(λq) by a cyclic group of order 2. Moreover, by considering a presentation PH̄(λq) of H̄(λq), we give the necessary and sufficient conditions of PH̄(λq) to be efficient on the minimal number of generators.