On the Residual Finiteness Growths of Particular Hyperbolic Manifold Groups

TL;DR

This paper provides explicit linear bounds on the residual finiteness growth of fundamental groups of certain hyperbolic manifolds, linking geometric properties with algebraic residual finiteness measures.

Contribution

It introduces a new linear upper bound on residual finiteness growth for hyperbolic manifolds with specific immersions, extending previous results to finite covers.

Findings

Linear bounds on residual finiteness growth in terms of geodesic length

Equivalence of geodesic residual finiteness growth and word length growth

Extension of bounds to finite covers with specific immersions

Abstract

We give a quantification of residual finiteness for the fundamental groups of hyperbolic manifolds that admit a totally geodesic immersion to a compact, right-angled Coxeter orbifold of dimension 3 or 4. Specifically, we give explicit upper bounds on residual finiteness that are linear in terms of geodesic length. We then extend the linear upper bounds to hyperbolic manifolds with a finite cover that admits such an immersion. Since the quantifications are given in terms of geodesic length, we define the geodesic residual finiteness growth and show that this growth is equivalent to the usual residual finiteness growth defined in terms of word length. This equivalence implies that our results recover the quantification of residual finiteness from \cite{BHP} for hyperbolic manifolds that virtually immerse into a compact reflection orbifold.