Group actions, Teichm\"uller spaces and cobordisms

TL;DR

This paper explores how the geometry and topology of 4-manifolds with hyperbolic structures relate to group actions of their fundamental groups, introducing unusual lattice actions on wild spheres and knots, linked to non-standard representation varieties.

Contribution

It presents novel constructions of unusual group actions on spheres and knots, connecting these actions to non-standard components of hyperbolic lattice representation varieties.

Findings

Unusual lattice actions correspond to non-standard representation components.

Constructed actions on wild spheres and knots exhibit hyperbolic geometric structures.

Demonstrated the relation between group actions and manifold topology in hyperbolic cobordisms.

Abstract

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).