*Geometric and algebraic structures on spaces of
quasiconformal mappings*

**Abstract:**

In the first part of the talk I will speak about the recent theorem which states the classification of biholomorphic and isometric mappings between Teichmuller spaces of arbitrary Riemann surfaces. In particular, I will discuss the new proof of the corresponding theorem of Royden for closed surfaces (joint with C. Earle).

In the second part I will talk about some interesting algebraic properties of the groups QS(S) and QC(D), that is the groups of quasisymmetric and quasiconformal mappings that act on the unit circle and the unit disc respectively. We show that there is no homomorphism from QS(S) to QC(D) which is splitting. Furthermore, we show that the group of normalized quasisymmetric mappings of the circle is not a simple group, while the group of normalized quasiconformal mappings of the two sphere is (joint with D. Epstein).