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Coning off totally geodesic boundary components of a hyperbolic manifold

Colby Kelln, Jason Manning

TL;DR

The paper develops explicit geometric conditions under which coning off the totally geodesic boundary components of a compact hyperbolic manifold yields a negatively curved cone-off. Central to the construction is a warped-product framework near the boundary, controlled by buffer width and boundary injectivity radius, and an Alexander–Bishop curvature bound applied to cone warping functions. The authors prove that, under suitable inequalities (and after passing to finite covers when needed), the cone-off carries a CAT(k)–type metric with negative curvature and that certain locally convex subspaces remain locally convex in the cone-off, which in turn implies injectivity of fundamental groups and quasi-convexity of subgroups. The work connects sharp geometric hypotheses to strong group-theoretic conclusions, including hyperbolicity of the cone-off's fundamental group and quasi-convexity of subspace images, enriching the toolkit for geometric Dehn-type fillings in hyperbolic geometry.

Abstract

Let M be a compact hyperbolic manifold with totally geodesic boundary. If the injectivity radius of the boundary is larger than an explicit function of the normal injectivity radius of the boundary, we show that there is a negatively curved metric on the space obtained by coning each boundary component of M to a point. Moreover, we give explicit geometric conditions under which a locally convex subset of M gives rise to a locally convex subset of the cone-off. Group-theoretically, we conclude that the fundamental group of the cone-off is hyperbolic and the image of the fundamental group of the coned-off locally convex subset is a quasi-convex subgroup.

Coning off totally geodesic boundary components of a hyperbolic manifold

TL;DR

The paper develops explicit geometric conditions under which coning off the totally geodesic boundary components of a compact hyperbolic manifold yields a negatively curved cone-off. Central to the construction is a warped-product framework near the boundary, controlled by buffer width and boundary injectivity radius, and an Alexander–Bishop curvature bound applied to cone warping functions. The authors prove that, under suitable inequalities (and after passing to finite covers when needed), the cone-off carries a CAT(k)–type metric with negative curvature and that certain locally convex subspaces remain locally convex in the cone-off, which in turn implies injectivity of fundamental groups and quasi-convexity of subgroups. The work connects sharp geometric hypotheses to strong group-theoretic conclusions, including hyperbolicity of the cone-off's fundamental group and quasi-convexity of subspace images, enriching the toolkit for geometric Dehn-type fillings in hyperbolic geometry.

Abstract

Let M be a compact hyperbolic manifold with totally geodesic boundary. If the injectivity radius of the boundary is larger than an explicit function of the normal injectivity radius of the boundary, we show that there is a negatively curved metric on the space obtained by coning each boundary component of M to a point. Moreover, we give explicit geometric conditions under which a locally convex subset of M gives rise to a locally convex subset of the cone-off. Group-theoretically, we conclude that the fundamental group of the cone-off is hyperbolic and the image of the fundamental group of the coned-off locally convex subset is a quasi-convex subgroup.
Paper Structure (9 sections, 19 theorems, 75 equations, 4 figures)

This paper contains 9 sections, 19 theorems, 75 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Statement of main theorems
  3. Preliminaries
  4. Warped products
  5. Local convexity
  6. Tailoring warped products to bound curvature from above
  7. Proof of Theorem \ref{['maintheorem']}
  8. Proof of Theorem \ref{['subspacetheorem']}
  9. The boundary at infinity of the coneoff.

Key Result

Theorem 1

Let $M$ be a compact hyperbolic manifold with totally geodesic boundary and suppose there are constants $b$ and $c$ so that and Then there is a negatively curved metric $\hat{d}$ on $\widehat{M}$ and a locally isometric embeddingHere "locally isometric" refers to the length metrics -- there is an underlying Riemannian metric for which this restriction is a Riemannian isometry (and hence a local

Figures (4)

  • Figure 1: An example $f(t)$ in pink and purple solid lines with tangent lines in dashed green and orange.
  • Figure 2: In the middle is a collar neighborhood of width less than $\operatorname{BW}_{M}(\partial M)$ of a component $N$ of $\partial M$. $\widehat{M}$ is on the left. On the right is the corresponding cone in $X$, with the removed portion of $M$ in dashed grey.
  • Figure 3: See Figure \ref{['fig:cone1']} for descriptions of the objects on the left and right. The pink brackets indicate $\phi_M$; the blue brackets indicate $\phi_C$.
  • Figure 4: A cartoon of $S\subset M$ (on the left) and $Y\subset X$ (on the right). The regions above level $b$ are Riemannian isometric.

Theorems & Definitions (55)

  • Definition 1.1
  • Definition 1.2
  • Example 1.3
  • Remark 1.4
  • Theorem 1
  • Remark 1.5
  • Corollary 1.6
  • proof
  • Definition 1.7: Projections of levels of $S$ to $\partial M$
  • Theorem 2
  • ...and 45 more