Group actions, Teichmüller spaces and cobordisms

Boris N. Apanasov

Published 2016-11-02Version 1

We discuss how the global geometry and topology of manifolds depend on different group actions of their fundamental groups, and in particular, how properties of a non-trivial compact 4-dimensional cobordism $M$ whose interior has a complete hyperbolic structure depend on properties of the variety of discrete representations of the fundamental group of its 3-dimensional boundary $\partial M$. In addition to the standard conformal ergodic action of a uniform hyperbolic lattice on the round sphere $S^{n-1}$ and its quasiconformal deformations in $S^n$, we present several constructions of unusual actions of such lattices on everywhere wild spheres (boundaries of quasisymmetric embeddings of the closed $n$-ball into $S^n$), on non-trivial $(n-1)$-knots in $S^{n+1}$, as well as actions defining non-trivial compact cobordisms with complete hyperbolic structures in its interiors. We show that such unusual actions always correspond to discrete representations of a given hyperbolic lattice from "non-standard" components of its varieties of representations (faithful or with large kernels of defining homomorphisms).