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# Conjugacy Classes and Binary Quadratic Forms for the Hecke Groups

Published:2012-11-13
Printed: Sep 2013
• Giabao Hoang,
Department of Mathematics, Franklin & Marshall College, Lancaster, PA 17604
• Wendell Ressler,
Department of Mathematics, Franklin & Marshall College, Lancaster, PA 17604
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## Abstract

In this paper we give a lower bound with respect to block length for the trace of non-elliptic conjugacy classes of the Hecke groups. One consequence of our bound is that there are finitely many conjugacy classes of a given trace in any Hecke group. We show that another consequence of our bound is that class numbers are finite for related hyperbolic $\mathbb{Z}[\lambda]$-binary quadratic forms. We give canonical class representatives and calculate class numbers for some classes of hyperbolic $\mathbb{Z}[\lambda]$-binary quadratic forms.
 Keywords: Hecke groups, conjugacy class, quadratic forms
 MSC Classifications: 11F06 - Structure of modular groups and generalizations; arithmetic groups [See also 20H05, 20H10, 22E40] 11E16 - General binary quadratic forms 11A55 - Continued fractions {For approximation results, see 11J70} [See also 11K50, 30B70, 40A15]