Geodesic Planes in Geometrically Finite Manifolds

TL;DR

This paper investigates the rigidity and density properties of geodesic plane immersions in geometrically finite manifolds with rank 1 cusps, revealing new criteria for non-closed immersions and properties of closed surfaces.

Contribution

It introduces new density criteria for geodesic plane immersions in geometrically finite manifolds with circle packing limit sets, and analyzes the properties of closed immersions and their fundamental groups.

Findings

K-thick recurrence of horocycles fails generically

Derived 2-density criteria for non-closed geodesic plane immersions

Closed immersions correspond to surfaces with finitely generated fundamental groups

Abstract

We study the problem of rigidity of closures of totally geodesic plane immersions in geometrically finite manifolds containing rank $1$ cusps. We show that the key notion of K-thick recurrence of horocycles fails generically in this setting. This property was introduced in the recent work of McMullen, Mohammadi and Oh. Nonetheless, in the setting of geometrically finite groups whose limit sets are circle packings, we derive 2 density criteria for non-closed geodesic plane immersions, and show that closed immersions give rise to surfaces with finitely generated fundamental groups. We also obtain results on the existence and isolation of proper closed immersions of elementary surfaces.