Nikolai V. Ivanov

Mapping Class Groups

The mapping class group Mod S of an orientable surface S is defined as the group of isotopy classes of orientation-preserving diffeomorphisms S → S . In addition to being a central object of the topology of surfaces (cf. 2.1), these groups also play an important role in the theory of Teichmuller spaces and in algebraic geometry, where they are known underthe name Teichmuller modular groups or simply modular groups . Our notations are derived from the latter terminology. There are several closely related groups, which also deserve the name of the mapping classgroups (or Teichmuller modular groups). First of all, one may consider the extended mapping class group Mod ◊ S of S defined as the group of the isotopy classes of all diffeomorphisms S → S . The pure mapping class group PMod S of S is defined as the group of isotopy classes of all orientation-preserving diffeomorphisms S → S preserving setwise all boundary components of S . Finally, one may consider the group ℳ S of all (orientation-preserving) diffeomorphisms S → S fixed on the boundary ∂ S , considered up to isotopies fixed on the boundary. If ∂ S ≠ 0, then diffeomorphisms fixed on ∂ S are automatically orientation-preserving. If ∂ S = 0, then, of course, ℳ S = Mod S . All these groups could be also defined as the 0- th homotopy groups of suitable diffeomorphisms groups of S . For example, Mod S = π 0 (Diff( S )), where Diff( S ) is the group of all orientation-preserving diffeomorphisms of S considered with, for example, C ∞ -topology (or any other reasonable topology).

Automorphisms of Complexes of Curves and of Teichmüller Spaces

To every compact orientable surface one can associate following Harvey Ha Ha a combinatorial object the so called complex of curves which is analogous to Tits buildings associated to semisimple Lie groups The basic result of the present paper is an analogue of a fundamental theorem of Tits for these complexes It asserts that every automorphism of the complex of curves of a surface is induced by some element of the Teichm uller modular group of this surface or what is the same by some di eomorphism of the surface in question This theorem allows us to give a completely new proof of a famous theorem of Royden R about isometries of the Teichm uller space In contrast with Royden s proof which is local and analytic this new proof is a global and geometric one and reveals a deep analogy between Royden s theorem and the Mostow s rigidity theorem Mo Mo Another application of our basic theorem is a complete description of isomorphisms between subgroups of nite index of a Teichm uller modular group This result in its turn has some further applications to modular groups

Teichmüller modular groups and arithmetic groups

In this paper we give a negative answer to a question of W. Harvey about the arithmeticity of Teichmüller modular forms and we prove a number of results about the nonarithmeticity of subnormal subgroups of modular groups. In the Appendix we announce and discuss a theorem according to which the appropriately defined rank of Teichmüller modular groups is equal to 1.

Arnol'd, the Jacobi Identity, and Orthocenters

Abstract The three altitudes of a plane triangle pass through a single point, called the orthocenter of the triangle. This property holds literally in Euclidean geometry, and, properly interpreted, also in hyperbolic and spherical geometries. Recently, V. I. Arnol’d offered a fresh look at this circle of ideas and connected it with the well-known Jacobi identity. The main goal of this article is to present an elementary version of Arnol’ds approach. In addition, several related ideas, including ones of M. Chasles, W. Fenchel and T. Jørgensen, and A. A. Kirillov, are discussed.