Topology of leaves for minimal laminations by hyperbolic surfaces

TL;DR

This paper constructs minimal laminations by hyperbolic surfaces with prescribed topologies, using towers of finite coverings and a novel relative residual finiteness approach to control surface properties.

Contribution

Introduces a relative residual finiteness method for constructing hyperbolic surface laminations with specific topological features.

Findings

Constructed minimal laminations with prescribed surface topologies

Developed a relative residual finiteness technique for surface covers

Controlled the second systole in finite surface covers

Abstract

We construct minimal laminations by hyperbolic surfaces whose generic leaf is a disk and contain any prescribed family of surfaces and with a precise control of the topologies of the surfaces that appear. The laminations are constructed via towers of finite coverings of surfaces for which we need to develop a relative version of residual finiteness which may be of independent interest. The main step in establishing this relative version of residual finiteness is to obtain finite covers with control on the \emph{second systole} of the surface, which is done in the appendix. In a companion paper, the case of other generic leaves is treated.