| Résume | An element A in PSL_2(\Z) is hyperbolic if |Tr(A)|>2. The maximal virtually abelian subgroup of PSL2(\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 surds). Doing so we revisit results of P. Sarnak (2007) , themselves rooted in classical work by Gauss and Fricke & Klein. This is joint work with C.-L. Simon (Rennes). |
