Groups and Dynamics Seminar

Date: April 30, 2025

Time: 2:00PM - 3:00PM

Location: BLOC 628

Speaker: Alain Valette, Université de Neuchâtel

Title: Reciprocal hyperbolic elements in PSL_2(Z)

Abstract: An element A in PSL_2(Z) is hyperbolic if |Tr(A)|>2. The maximal virtually abelian subgroup of PSL_2(Z) containing A is either infinite cyclic or infinite dihedral; say that A is reciprocal if the second case happens (A is then conjugate to its inverse). We give a characterization of reciprocal hyperbolic elements in PSL_2(Z) in terms of the continued fractions of their fixed points in P^1(R) (those are quadratic irrationals). Doing so we revisit results of P. Sarnak (2007) and C.-L. Simon (2022), themselves rooted in classical work by Gauss and Fricke & Klein.