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Peter B. Shalen

Publications and source records attributed to Peter B. Shalen.

At least 19 recordsLinked to original sources
arXiv2026

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

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 instance, if vol(\orbM) < 0.1491, then the underlying space of \orbM must be either a small Seifert fibered space, or the connected sum of two lens spaces (or of a lens space with S^2 x S^1), or the gluing of one or two highly restricted Seifert fibered spaces along a single incompressible torus. If vol(\orbM) < 0.1571 and the singular locus of \orbM is a link, then the topology of the underlying space is restricted even further. Our methods are primarily topological, and involve studying the underlying topology of orbifold books of I-bundles. Along the way, we provide an exposition of some foundational material about orbifolds that was previously absent from the literature.

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arXiv2025

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

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 is greater than $2V_{\rm oct}$, where $V_{\rm oct}=3.66\ldots$ denotes the volume of a regular ideal hyperbolic octahedron in $\mathbb{H}^3$. This improves the lower bound given in our earlier paper ``Volume and topology of bounded and closed hyperbolic $3$-manifolds.'' One ingredient in the improved bound is that in a crucial case, instead of using a single ``muffin'' in $N$ in the sense of Kojima and Miyamoto, we use two disjoint muffins. By combining the result about manifolds with geodesic boundary with the $\log(2k-1)$ theorem and results due to Agol-Culler-Shalen and Shalen-Wagreich, we show that if $M$ is a closed, orientable hyperbolic $3$-manifold with $\mathop{\rm vol} M\le V_{\rm oct}/2$, then $\dim H_1(M;\mathbb{F}_2)\le4$. We also provide new lower bounds for the volumes of closed hyperbolic $3$-manifolds whose cohomology ring over $\mathbb{F}_2$ satisfies certain restrictions; these improve results that were proved in ``Volume and topology$\ldots$.''

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arXiv2025

Actualizing subgroups of 3-manifold groups in homologically small submanifolds

Let $Y$ be a simple $3$-manifold, and let $A$ be a finitely generated, freely indecomposable subgroup of $π_1(Y)$. Set $η=\dim H_1(A;{\bf F}_2)$. Suppose that either (a) $\partial Y\ne\emptyset$ or (b) $\dim H_1(Y;{\bf F}_2)\ge3η^2-4η+4$. Under these hypotheses, we show that $A$ is carried by some compact, connected three-dimensional submanifold $Z$ of $\text{int} \;Y$ such that (1) $\partial Z$ is non-empty, and each of its components is incompressible in $Y$; (2) the Euler characteristic of $Z$ is bounded below by $1-η$; and (3) $\dim H_1(Z;{\bf F}_2)\le 3η^2-4η+1$. The conclusion implies that any boundary component of $Z$ is an incompressible surface of genus at most $η$. In Case (b), this should be compared with earlier results proved by Agol-Culler-Shalen and Culler-Shalen, which provide a surface of genus at most $η$ under weaker hypotheses (the lower bound on $\dim H_1(Y; {\bf F}_2)$ being linear in $η$ rather than quadratic), but do not give any relationship between the given subgroup $A$ and this surface. In a forthcoming paper we will apply the result to give a new upper bound for the ratio of the rank of the mod 2 homology of a closed, orientable hyperbolic $3$-manifold to the volume of the manifold.

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arXiv2023

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

Let $p$ be a point of an orientable hyperbolic $3$-manifold $M$, and let $m\ge1$ and $k\ge2$ be integers. Suppose that $α_1,\ldots,α_m$ are loops based at $p$ having length less than $\log(2k-1)$. We show that if $G$ denotes the subgroup of $π_1(M,p)$ generated by $[α_1],\ldots,[α_m]$, then $\overlineχ(G)\doteq-χ(G)\le k-2$; here $χ(G)$ denotes the Euler characteristic of the group $G$, which is always defined in this situation. This result is deduced from a result about an arbitrary finitely generated subgroup $G$ of the fundamental group of an orientable hyperbolic $3$-manifold. If $Δ$ is a finite generating set for $G$, we define the $index\ of\ freedom$ ${\rm iof}(Δ)$ to be the largest integer $k$ such that $Δ$ contains $k$ elements that freely generate a rank-$k$ free subgroup of $G$. We define the $minimum\ index\ of\ freedom$ ${\rm miof}(G)$ to be $\min_{Δ}{\rm iof}(Δ)$, where $Δ$ ranges over all finite generating sets for $G$. The result is that $\overlineχ(G)<{\rm iof}(G)$. The author has recently learned that this is equivalent to a special case of a theorem about arbitrary finitely presented groups due to J. S. Wilson.

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arXiv2022

The ratio of homology rank to hyperbolic volume, I

We show that for every finite-volume hyperbolic $3$-manifold $M$ and every prime $p$ we have $\text{dim}\ H_1(M;\mathbf{F}_p)< 168.602\cdot\text{vol}\ M$. There are slightly stronger estimates if $p = 2$ or if $M$ is non-compact. This improves on a result proved by Agol, Leininger and Margalit, which gave the same inequality with a coefficient of $334.08$ in place of $168.602$. It also improves on the analogous result with a coefficient of about $260$, which could have been obtained by combining the arguments due to Agol, Leininger and Margalit with a result due to Böröczky. Our inequality involving homology rank is deduced from a result about the rank of the fundamental group: if $M$ is a finite-volume orientable hyperbolic $3$-manifold such that $π_1(M)$ is $2$-semifree, then $\text{rank}\ π_1(M)<1+λ_{0}\cdot\text{vol}\ M$, where $λ_{0}$ is a certain constant less than $167.79$

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arXiv2022

The ratio of homology rank to hyperbolic volume, II

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 $3$ manifold $M$ in terms of its volume. A surprising feature of the arguments in the paper is that they require an application of the Four Color Theorem. If $M$ is closed, and either (a) $π_1(M)$ has no subgroup isomorphic to the fundamental group of a closed, orientable surface of genus $2, 3$ or $4$, or (b) $p = 2$, and $M$ contains no (embedded, two-sided) incompressible surface of genus $2, 3$ or $4$, then $\text{dim}\, H_1(M;F_p) < 157.763 \cdot \text{vol}(M)$. If $M$ has one or more cusps, we get a very similar bound assuming that $π_1(M)$ has no subgroup isomorphic to the fundamental group of a closed, orientable surface of genus $g$ for $g = 2, \dots,8$. These results should be compared with those of our previous paper $The\ ratio\ of\ homology\ rank\ to\ hyperbolic\ volume,\ I$, in which we obtained a bound with a coefficient in the range of $168$ instead of $158$, without a restriction on surface subgroups or incompressible surfaces. In a future paper, using a much more involved argument, we expect to obtain bounds close to those given by the present paper without such a restriction. The arguments also give new linear upper bounds (with constant terms) for the rank of $π_1(M)$ in terms of $\text{vol}\,M$, assuming that either $π_1(M)$ is $9$-free, or $M$ is closed and $π_1(M)$ is $5$-free.

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arXiv2021

Hyperbolic volume, mod 2 homology, and k-freeness

We show that if $M$ is any closed, orientable hyperbolic $3$-manifold with ${\rm vol}\ M\le3.69$, we have ${\rm dim}\ H_1(M;{\bf F}_2)\le7$. This may be regarded as a qualitative improvement of a result due to Culler and Shalen, because the constant $3.69$ is greater than the ordinal corresponding to $ω^2$ in the well-ordered set of finite volumes of hyperbolic $3$-manifolds. We also show that if ${\rm vol}\ M\le 3.77$, we have ${\rm dim}\ H_1(M;{\bf F}_2)\le10$. These results are applications of a new method for obtaining lower bounds for the volume of a closed, orientable hyperbolic $3$-manifold such that $π_1(M)$ is $k$-free for a given $k\ge4$. Among other applications we show that if $π_1(M)$ is $4$-free we have ${\rm vol}\ M>3.57$ (improving the lower bound of $3.44$ given by Culler and Shalen), and that if $π_1(M)$ is $5$-free we have ${\rm vol}\ M>3.77$.

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arXiv2020

Four-free groups and hyperbolic geometry

We give new information about the geometry of closed, orientable hyperbolic 3-manifolds with 4-free fundamental group. As an application we show that such a manifold has volume greater than 3.44. This is in turn used to show that if M is a closed orientable hyperbolic 3-manifold such that vol M < 3.44, then H_1(M;Z/2Z) has dimension at most 7.

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arXiv2019

Volume and Homology for Hyperbolic 3-Orbifolds

Let ${\mathfrak M}$ be a closed, orientable, hyperbolic 3-orbifold such that $π_1({\mathfrak M})$ contains no hyperbolic triangle group. We show that strict upper bounds of 0.07625, 0.1525 and 0.22875 for ${\rm vol}\ {\mathfrak M}$ imply respective upper bounds of 23, 43 and 79 for $\dim H_1({\mathfrak M};{\mathbb F}_2 )$. Stronger results hold if we assume that the singular set $Σ$ is a link; specifically, under this assumption, strict upper bounds of 0.305, 0.4575, 0.61, 0.7625 and 0.915 for ${\rm vol}\ {\mathfrak M}$ imply respective upper bounds of 7, 13, 14, 28 and 29 for ${\rm dim}\ H_1({\mathfrak M};{\mathbb F}_2 )$. Irreducibility assumptions on the underlying manifold $|{\mathfrak M}|$ of ${\mathfrak M}$, and of the underlying manifolds of certain coverings of ${\mathfrak M}$, also give stronger results. The upper bounds on $\dim H_1({\mathfrak M};{\mathbb F}_2 )$ for an orbifold ${\mathfrak M}$ whose volume is subject to a suitable upper bound are deduced from upper bounds on ${\rm dim}\ H_1(|{\mathfrak M}|;{\mathbb F}_2 )$ for an orbifold ${\mathfrak M}$ whose volume is subject to a suitable upper bound.

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arXiv2018

The geometry of $k$-free hyperbolic $3$-manifolds

We investigate the geometry of closed, orientable, hyperbolic $3$-manifolds whose fundamental groups are $k$-free for a given integer $k\ge 3$. We show that any such manifold $M$ contains a point $P$ of $M$ with the following property: If $S$ is the set of elements of $π_1(M,P)$ represented by loops of length $<\log(2k-1)$, then for every subset $T \subset S$, we have ${\rm rank}\ T \le k-3$. This generalizes to all $k\ge3$ results proved in [6] and [10], which have been used to relate the volume of a hyperbolic manifold to its topological properties, and it strictly improves on the result obtained in [11] for $k=5$. The proof avoids the use of results about ranks of joins and intersections in free groups that were used in [10] and [11].

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arXiv2017

Volume and Homology for Hyperbolic $3$-Orbifolds, I

Let ${\mathfrak M}$ be a closed, orientable, hyperbolic 3-orbifold whose singular set is a link, and such that $π_1({\mathfrak M})$ contains no hyperbolic triangle group. We show that if the underlying manifold $|{\mathfrak M}|$ is irreducible, and $|{\mathfrak M}|$ is irreducible for every two-sheeted (orbifold) covering $\widetilde{\mathfrak M}$ of ${\mathfrak M}$, and if ${\rm vol} {\mathfrak M}\le1.72$, then $\dim H_1({\mathfrak M};{\mathbb Z}_2)\le 15$. Furthermore, if ${\rm vol} {\mathfrak M}\le1.22$ then $\dim H_1({\mathfrak M};{\mathbb Z}_2)\le 11$, and if ${\rm vol} {\mathfrak M}\le0.61$ then $\dim H_1({\mathfrak M};{\mathbb Z}_2)\le 7$. The proof is an application of results that will be used in the sequel to this paper to obtain qualitatively similar results without the assumption of irreducibility of $|{\mathfrak M}|$ and $|\widetilde{\mathfrak M}|$.

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arXiv2011

Orders of elements in finite quotients of Kleinian groups

A positive integer $m$ will be called a {\it finitistic order} for an element $γ$ of a group $Γ$ if there exist a finite group $G$ and a homomorphism $h:Γ\to G$ such that $h(γ)$ has order $m$ in $G$. It is shown that up to conjugacy, all but finitely many elements of a given finitely generated, torsion-free Kleinian group admit a given integer $m>2$ as a finitistic order.

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arXiv2011

Margulis numbers and number fields

It is shown that, up to isometry, all but finitely many closed, orientable hyperbolic 3-manifolds with a given trace field $K$ admit 0.34 as a Margulis number. This is deduced from a more technical result giving a condition under which $\max(d(P,x\cdot P),d(P,y\cdot P))\ge0.34$ for every $P\in\HH^3$, where $x$ and $y$ lie in $\pizzle(E)$ for some number field $E$, generate a discrete torsion-free group of $\pizzle(\CC)$ and do not commute. Specifically, this is always the case if there is a valuation $v$ of $E$ such that (1) the residue field $k_v=\frako_v/\frakm_v$ of $v$ has sufficiently large characteristic, (2) $x\in\pizzle(\frako_v)$, and (3) the image of $x$ under the natural homomorphism $\pizzle(\frako_v)\to \pizzle(k_v)$ has order 7.

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arXiv2011

Margulis numbers for Haken manifolds

For every closed hyperbolic Haken 3-manifold and, more generally, for any hyperbolic 3-manifold M which is homeomorphic to the interior of a Haken manifold, the number 0.286 is a Margulis number. If M has non-zero first Betti number, or if M is closed and contains a semi-fiber, then 0.292 is a Margulis number for M.

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arXiv2010

Singular surfaces, mod 2 homology, and hyperbolic volume, II

If M is a closed simple 3-manifold whose fundamental group contains a genus-g surface group for some g>1, and if the dimension of H_1(M;Z_2) is at least max(3g-1,6), we show that M contains a closed, incompressible surface of genus at most g. This improves the main topological result of part I, in which the the same conclusion was obtained under the stronger hypothesis that the dimension of H_1(M;Z_2) is at least 4g-1. As an application we show that if M is a closed orientable hyperbolic 3-manifold with volume at most 3.08, then H_1(M;Z_2) has dimension at most 5.

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arXiv2010

Small optimal Margulis numbers force upper volume bounds

If $λ$ is a positive real number strictly less than $\log3$, there is a positive number $V_λ$ such that every orientable hyperbolic 3-manifold of volume greater than $V_λ$ admits $λ$ as a Margulis number. If $λ<(\log3)/2$, such a $V_λ$ can be specified explicitly, and is bounded above by $$λ\bigg(6+\frac{880}{\log3-2λ}\log{1\over\log3-2λ}\bigg),$$ where $\log$ denotes the natural logarithm. These results imply that for $λ<\log3$, an orientable hyperbolic 3-manifold that does not have $λ$ as a Margulis number has a rank-2 subgroup of bounded index in its fundamental group, and in particular has a fundamental group of bounded rank. Again, the bounds in these corollaries can be made explicit if $λ<(\log3)/2$.

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arXiv2009

Volume and topology of bounded and closed hyperbolic 3-manifolds

Let N be a compact, orientable hyperbolic 3-manifold with connected, totally geodesic boundary of genus 2. If N has Heegaard genus at least 5, then its volume is greater than 6.89. The proof of this result uses the following dichotomy: either N has a long return path (defined by Kojima-Miyamoto), or N has an embedded, codimension-0 submanifold X with incompressible boundary $T \sqcup \partial N$, where T is the frontier of X in N, which is not a book of I-bundles. As an application of this result, we show that if M is a closed, orientable hyperbolic 3-manifold such that H_1(M;Z_2) has dimension at least 5, and if the image in H^2(M;Z_2) of the cup product map has image of dimension at most 1, then M has volume greater than 3.44.

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