Abstracts of Nikolai V. Ivanov's Preprints

  1. Arnol'd, Jacobi identity, and orthocenters

    V.I. Arnol'd had recently discovered that the Jacobi identity lies at the heart of the theory of altitudes of a triangle in hyperbolic geometry. Arnol'd's approach is based on the Jacobi identity for the Poisson bracket of the quadratic forms on a symplectic plane. The goal of this article is to present an elementary approach to these ideas of Arnol'd. It is based on the most simple form of the Jacobi identity, namely, the Jacobi identity for the commutator [A,B]=AB-BA of 2-by-2 matrices. We also give a proof of the altitudes theorem in the spherical geometry based on the Jacobi identity for the usual vector product, and discuss the Euclidean case.

    We assume that the reader is familiar with the Poincaré unit disc model of the hyperbolic geometry, and with the projective plane. Beyond this, only the first notions of the linear algebra are needed.

  1. The Lax proof of the change of variables formula, differential forms, a determinantal identity, and Jacobi multipliers

    The first goal of this note is to present a differential forms version of a beautiful proof by P. Lax of (a special case of) the change of variables formula. Our second goal is to present a fairly detailed comparison of our proof with the Lax one. Such a comparison is very instructive; it sheds a light on both the efficiency of the differential form theory and the brilliance with which Lax uses the classical analysis. A key role in the Lax proof is played by a fairly mysterious determinantal identity. Our third goal is to trace the roots of this identity. Its explanation from the point of view of differential forms was provided in my paper A topologists view of the Dunford-Schwartz proof of the Brouwer fixed point theorem (No. 39 in the list of publications).

  2. Isometries of Teichmüler spaces from the point of view of Mostow rigidity

    The purpose of this paper is to present a new proof of the famous theorem of Royden and Earle-Kra to the effect that all isometries of Teichmüller spaces actually belong to Teichmüller modular groups. This new proof uncovers a deep and unexpected analogy of this theorem with the Mostow rigidity theorem for lattices in semisimple Lie groups of rank at least 2.

  3. Mapping class groups

    This is a research-expository paper for the Handbook in Geometric Topology, edited by R. Daverman and R. Sher. The paper is centered around three topics: generators and relations of the mapping class groups; cohomology propertries of the mapping class groups; automorphisms of complexes of curves and their applications. For the first two topics, many detailed proofs are included, some of them new. Complexes of curves serve as a unifying thread for the whole paper.

  4. A topologists view of the Dunford-Schwartz proof of the Brouwer fixed point theorem

    This paper was provoked by some remarks of G.-C. Rota in his recent remarkable book Indiscete Thoughts. It explains that the analytical proof of the Brouwer fixed point theorem presented in the famous treatise of Dunford and Schwartz is essentially the usual topological proof in a disguise. The main tools used to uncover this are differential forms and de Rham cohomology.

  5. A short proof of Gromov non-hyperbolicity of Teichmüller spaces

    This new proof of a theorem of H. Masur and M. Wolf is based on the observation that the pairs of geodesic rays in Teichmüller space constructed by H. Masur in his 1975 Annals of Mathematics paper are not exponentially divergent. This is not difficult to see, but one should pay special attention to the specific form of the definition of exponential divergence in the theory of Gromov hyperbolic spaces.

  6. (with John D. McCarthy) On injective homomorphisms between Teichmüller modular groups

    In this paper we prove that injective homomorphisms between Teichmüller modular groups of compact orientable surfaces are necessary isomorphisms, if an appropriately measured "size" of the surfaces in question differs by at most one. In particular, we establish the co-Hopfian property for modular groups of surfaces of positive genus.

    For the publication the paper was revised and split into two, the first one dealing with the generic case of genus at least two, and the second one dealing with case where one of the surfaces has genus one. Also, some expository material was deleted. The first part is the paper No. 38 in the publication lists. Here is the original version.

  7. A geometric interpretation of the complex dilatation

    The purpose of this paper is to provide a geometric interpretation of the familiar formula defining the complex dilatation of a map. The main observation is that the differential of a map at a point naturally defines a point in the projective model of the hyperbolic plane, and that we will get exactly the usual complex dilatation if we pass to the conformal disc model.

  8. Ten Problems on the Mapping Class Groups

    In early 1994 I prepared a list of problems about the mapping class groups, stimulated largely by the request of R. Kirby to submit some problems for the new version of his famous list. Some of my problems ended up in the Kirby's list (of course, in a seriously edited form), some did not. Here is the original list with several updates reflecting some later developments, with the last update in December 1997. This note is preserved here as a historical document.