Jason DeBlois

Totally geodesic surfaces in hyperbolic 3-manifolds

v List of Figures viii Chapter

Some virtually special hyperbolic 3-manifold groups

Let M be a complete hyperbolic 3-manifold of finite volume that admits a decomposition into right-angled ideal polyhedra. We show that M has a deformation retraction that is a virtually special square complex, in the sense of Haglund and Wise and deduce that such manifolds are virtually fibered. We generalise a theorem of Haglund and Wise to the relatively hyperbolic setting and deduce that the fundamental group of M is LERF and that the geometrically finite subgroups of the fundamental group are virtual retracts. Examples of 3-manifolds admitting such a decomposition include augmented link complements. We classify the low-complexity augmented links and describe an infinite family with complements not commensurable to any 3-dimensional reflection orbifold.

Algebraic invariants, mutation, and commensurability of link complements

We construct a family of hyperbolic link complements by gluing tangles along totally geodesic four-punctured spheres, then investigate the commensurability relation among its members. Those with different volume are incommensurable, distinguished by their scissors congruence classes. Mutation produces arbitrarily large finite subfamilies of nonisometric manifolds with the same volume and scissors congruence class. Depending on the choice of mutation, these manifolds may be commensurable or incommensurable, distinguished in the latter case by cusp parameters. All have trace field Q(i,\sqrt{2}), but some have integral traces while others do not.

The centered dual and the maximal injectivity radius of hyperbolic surfaces

We give sharp upper bounds on the maximal injectivity radius of finite-area hyperbolic surfaces and use them, for each g 2, to identify a constant rg 1;2 such that the set of closed genus-g hyperbolic surfaces with maximal injectivity radius at least r is compact if and only if r > rg 1;2 . The main tool is a version of the centered dual complex that we introduced earlier, a coarsening of the Delaunay complex. In particular, we bound the area of a compact centered dual two-cell below given lower bounds on its side lengths. 52C15, 57M50 This paper analyzes the centered dual complex of a locally finite subset S of H 2 , first introduced in our prior preprint [6], and applies it to describe the maximal injectivity radius of hyperbolic surfaces. The centered dual complex is a cell decomposition with vertex set S and totally geodesic edges. Its underlying space contains that of the geometric dual to the Voronoi tessellation. We regard it as a tool for understanding the geometry of packings. The rough idea behind the construction is that geometric dual 2‐cells that are not centered (see Definition 0.2) are hard to analyze individually but naturally group into larger cells that can be treated as units. Our first main theorem bears the fruit of this approach, turning a lower bound on edge lengths into a good lower bound on area for centered dual 2‐cells.

The Delaunay tessellation in hyperbolic space

The Delaunay tessellation of a locally finite subset of hyperbolic space is constructed using convex hulls in Euclidean space of one higher dimension. For finite and lattice-invariant sets it is proven to be a polyhedral decomposition, and versions (necessarily modified from the Euclidean setting) of the empty circumspheres condition and geometric duality with the Voronoi tessellation are proved. Some pathological examples of infinite, non lattice-invariant sets are exhibited.

Surface groups are frequently faithful

We show the set of faithful representations of a closed orientable hyperbolic surface group is dense in both irreducible components of the PSL2(K) representation variety, where K = C or R, answering a question of W. Goldman. We also prove the existence of faithful representations into PU(2, 1) with certain nonintegral Toledo invariants.

Explicit rank bounds for cyclic covers

Let

Cross curvature flow on a negatively curved solid torus

M

Totally geodesic surfaces and homology

be a closed, orientable hyperbolic 3-manifold and

The geometry of cyclic hyperbolic polygons

\phi