Stochastic processes on geometric loop and diffeomorphism groups of real and complex manifolds, associated unitary representations

Semidirect products of groups of loops and groups of diffeomorphisms of real, complex and quaternion manifolds

This article is devoted to the investigation of semidirect products of groups of loops and groups of diffeomorphisms of finite and infinte dimensional real, complex and quaternion manifolds. Necessary statements about quaternion manifolds with quaternion holomorphic transition mappings between charts of atlases are proved. Unitary representations of these groups including irreducible are constructed with the help of stochastic processes and quasi-invariant transition measures on groups $G$ relative to dense subgroups $G'$. It is proved, that this procedure provides a family of the cardinality …

Groups of diffeomorphisms and geometric loops of manifolds over ultra-normed fields

The article is devoted to the investigation of groups of diffeomorphisms and loops of manifolds over ultra-metric fields of zero and positive characteristics. Different types of topologies are considered on groups of loops and diffeomorphisms relative to which they are generalized Lie groups or topological groups. Among such topologies pairwise incomparable are found as well. Topological perfectness of the diffeomorphism group relative to certain topologies is studied. There are proved theorems about projective limit decompositions of these groups and their compactifications for compact manifo…

Profinite and Finite Groups Associated with Loop and Diffeomorphism Groups of Non-Archimedean Manifolds

$p$-Adic compactifications of geometric loop and diffeomorphism groups of compact manifolds on finite-dimensional spaces over non-Archimedean fields are investigated. Weakened topology is introduced. The structure of newly constructed compact groups is studied. Representations of such profinite groups are discused.

Anosov diffeomorphisms on Thurston geometric 4-manifolds

A long-standing conjecture asserts that any Anosov diffeomorphism of a closed manifold is finitely covered by a diffeomorphism which is topologically conjugate to a hyperbolic automorphism of a nilpotent manifold. In this paper, we show that any closed 4-manifold that carries a Thurston geometry and is not finitely covered by a product of two aspherical surfaces does not support (transitive) Anosov diffeomorphisms.

简单多胞形的微分同胚的边界值，以及可控性

Boundary values of diffeomorphisms of simple polytopes, and controllability

本文研究简单多胞形上微分同胚群的边界值问题，建立了商空间的典范李群结构，并证明恒等分量由指数映射生成，得出可控性结论。

Conjugation-invariant norms on groups of geometric origin

A group is said to be bounded if it has a finite diameter with respect to any bi-invariant metric. In the present paper we discuss boundedness of various groups of diffeomorphisms.