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
 the 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.