The lecture is about the so-called inverse spectral geometry in the domain of Fuchsian groups, or what is the same, Riemann surfaces endowed with a hyperbolic metric. It is well known that two Fuchsian groups of a given signature (g,m) have the same spectrum of the Laplacian if and only if they have the same spectrum of the traces of the elements of the group (trace spectrum). The general question is, to what extent does either spectrum determine the group.

For signatures not too small (e.g. all signatures (g, 0) with g >3), isospectral non conjugate examples are known. The lecture will be about methods proving that in the case of small signatures the group is entirely determined by the spectrum.