Speaker
Alain Valette
(Université de Neuchâtel)
Description
An element $A$ in $\operatorname{PSL}_2(\mathbb{Z})$ is hyperbolic if $|\operatorname{Tr}(A)|>2$. The maximal virtually abelian subgroup of $\operatorname{PSL}_2(\mathbb{Z})$ 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 $\operatorname{PSL}_2(\mathbb{Z})$ in terms of the continued fractions of their fixed points in $\operatorname{P}^1(\mathbb{R})$ (those are quadratic surds). Doing so we revisit results of P. Sarnak (2007) and C.-L. Simon (2022), themselves rooted in classical work by Gauss and Fricke \& Klein.
https://sites.google.com/view/julgfest