P. Shalen

activity

20022026

most citedDehn surgery, homology and hyperbolic volume

34 citations · 86 across the 25 of their papers we have counts for

collaborators

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25 papers · 1 filter

math.GT2026

The underlying manifold of a low-volume hyperbolic 3-orbifold

David Futer, Peter B. Shalen

This paper proves strong restrictions on the underlying topological space of a closed, orientable hyperbolic 3-orbifold \orbM whose volume satisfies a certain upper bound. For inst…

math.GT2024

Volume and topology of bounded and closed hyperbolic 3-manifolds, II

Jason DeBlois, Peter B. Shalen

Let $N$ be a compact, orientable hyperbolic 3-manifold whose boundary is a connected totally geodesic surface of genus $2$ . If $N$ has Heegaard genus at least $5$ , then its volume…

math.GT2023

Euler characteristics, lengths of loops in hyperbolic 3-manifolds, and Wilson's Freiheitssatz

Peter B. Shalen

Let $p$ be a point of an orientable hyperbolic $3$ -manifold $M$ , and let $m \geq 1$ and $k \geq 2$ be integers. Suppose that $α_{1}, \dots, α_{m}$ are loops based at $p$ having length less th…

math.GT2022

The ratio of homology rank to hyperbolic volume, II

Rosemary K. Guzman, Peter B. Shalen

Under mild topological restrictions, we obtain new linear upper bounds for the dimension of the mod $p$ homology (for any prime $p$ ) of a finite-volume orientable hyperbolic $3$ ma…

math.GT2021

The ratio of homology rank to hyperbolic volume, I

Rosemary K. Guzman, Peter B. Shalen

We show that for every finite-volume hyperbolic $3$ -manifold $M$ and every prime $p$ we have $dim H_{1} (M; F_{p}) < 168.602 \cdot vol M$ . There are slightly strong…

math.GT2020

Hyperbolic volume, mod 2 homology, and k-freeness

Rosemary K. Guzman, Peter B. Shalen

We show that if $M$ is any closed, orientable hyperbolic $3$ -manifold with $vol M \leq 3.69$ , we have $dim H_{1} (M; F_{2}) \leq 7$ . This may be regarded as a qualitative i…