arXiv subjects

Matthew Zevenbergen

Publications and source records attributed to Matthew Zevenbergen.

3 recordsLinked to original sources

arXiv2026

Combinations and chromatography of paths of Kleinian groups

We study continuous paths in the Chabauty topology on the set $\mathcal{D}_n$ of torsion free discrete subgroups of the isometry group of $n$-dimensional hyperbolic space. We prove a combination theorem for paths in $\mathcal{D}_n$, which allows us to construct an exotic path of discrete subgroups along which no two subgroups are isomorphic. We also introduce a technique we refer to as "chromatography" to prove a decomposition theorem that characterizes paths of convex cocompact groups in $\mathcal{D}_3$.

math.GT↗

arXiv2025

Systolic lattice extensions of classical Schottky groups

We produce lattice extensions of a dense family of classical Schottky subgroups of the isometry group of $d$-dimensional hyperbolic space. The extensions produced are said to be systolic, since all loxodromic elements with short translation length are conjugate into the Schottky groups. Various corollaries are obtained, in particular showing that for all $d\geq3$, the set of complex translation lengths realized by systoles of closed hyperbolic $d$-manifolds is dense inside the set of all possible complex translation lengths. We also consider complex translation lengths in arithmetic hyperbolic $d$-manifolds, and provide a new way to construct non-arithmetic lattices.

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arXiv2024

Connectivity in the space of framed hyperbolic 3-manifolds

We prove that the space $\mathcal{H}_\infty$ of framed infinite volume hyperbolic $3$-manifolds is connected but not path connected. Two proofs of connectivity of this space, which is equipped with the geometric topology, are given, each utilizing the density theorem for Kleinian groups. In particular, we construct a hyperbolic $3$-manifold whose set of framings is dense in $\mathcal{H}_\infty$. Examples of paths in $\mathcal{H}_\infty$ are discussed, including paths of geometrically finite manifolds limiting to certain infinite type geometric limits of quasi-Fuchsian manifolds. The discussion of paths culminates in describing an infinite family of non-tame hyperbolic $3$-manifolds, each of whose set of framings is a path component of $\mathcal{H}_\infty$, establishing that $\mathcal{H}_\infty$ is not path connected.

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