Abstract: 

This talk investigates symmetric (or reversible) conjugacy classes in the modular group, namely those consisting of elements that are conjugate to their inverses. These classes are intimately connected with classical objects from number theory, including periodic continued fractions and binary quadratic forms, as well as with closed geodesics on modular surfaces and aspects of symbolic dynamics. Motivated by Sarnak’s analytic classification of such classes via involutions, we develop a purely combinatorial framework to study them. Exploiting the free product structure of the modular group, we describe these conjugacy classes in terms of reduced words and their associated exponent sequences. Their symmetry is reflected in anti-periodicity and palindromic properties, which in turn allow for explicit enumeration and asymptotic counting results. This is joint work with Debattam Das and Khushi Mishra.

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