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UPDATE: Because I was hoping that state the question as concisely as possible, the original post did not include a precise definition of arithmetic 3-manifold only a reference to Maclachlan and Reid's book where it is defined. However, this lead to an ambiguity that caused conflicting answers. With the included definition, Ian Agol's answer is correct. However, given a more restrictive definition of arithmetic 3-manifold, which is (less standard, but) used in a paper of Gromov and Guth, John Pardon's answer is correct. I apologize for the confusion and appreciate the thoughtful responses of the MO community.

Following a definition was taken from Maclachlan and Reid, ``The Arithmetic of Hyperbolic 3-manifolds'', Chapter 8.2:

Let $k$ be a number field with exactly one complex place and $A$ a quaternion algebra over $k$ which is ramified at all real places. Furthermore assume $A\subset M_2(\mathbb{C})$ via a complex embedding of $k$. If $\mathcal{O}$ is an order in $A$ and $\mathcal{O}^1$ is the set of elements of $\mathcal{O}$ of determinant 1, then a subgroup $\Gamma$ of $SL(2,\mathbb{C})$ is arithmetic if it is commensurable with $\mathcal{O}^1$ and a subgroup $\Gamma \subset PSL(2,\mathbb{C})$ is arithmetic if it is commensurable with $P\mathcal{O}^1$. If we consider the natural action of $PSL(2,\mathbb{C})$ on the upper-half space model of $\mathbb{H}^3$, a finite volume hyperbolic 3-manifold (or 3-orbifold) $Q \cong \mathbb{H}^3/\Gamma$ is arithmetic precisely when $\Gamma$ is arithmetic.

Arithmetic 3-manifolds tend to be atypical many ways. For example, one non-generic property of arithmeticity observed by A.Borel in "Commensurability classes and volumes of hyperbolic 3-manifolds" is the following: for a fixed $V>0$, there are only finitely arithmetic 3-manifolds of volume less than $V$.

Along those lines, I was wondering about the following:

Question: For a fixed Heegaard genus $g$, are there only finitely many arithmetic manifolds of Heegaard genus $<g?$

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2 Answers 2

up vote 10 down vote accepted

The answer is no, for trivial reasons: if a hyperbolic 3-manifold fibers over $S^1$ with fiber of genus $g$, then the Heegaard genus is bounded by $2g+1$. However, cyclic covers of this manifold dual to the fiber will retain this property, so there are infinitely many manifolds of bounded Heegaard genus. One may obtain similar infinite families from semi-fiberings.

The error in Pardon's answer is that the Gromov-Guth estimate (or earlier estimate of Lackenby) only applies to congruence arithmetic manifolds (at least, Gromov-Guth seem to be missing this hypothesis).

On the other hand, this is the only way in which there can be an infinite family of arithmetic 3-manifolds of bounded Heegaard genus. The sketch of the argument is given an arithmetic 3-manifold of genus $g$, it will cover a minimal orbifold, which is congruence. Then one applies Lackenby's argument to show that the volume of the orbifold is bounded, and thus by Borel, such manifolds cover finitely many minimal orbifolds. Then one knows that for each such orbifold, all the covers with bounded genus (in fact bounded rank) come from covers of finitely many fiberings or semi-fiberings (see Biringer-Souto for (most of) the argument).

Conjecturally, this argument applies to manifolds of bounded rank, but so far is only known with an extra hypothesis of lower bound on the injectivity radius, which would follows from Lehmer's conjecture (see again Biringer-Souto). I've made many attempts at proving this without Lehmer's conjecture, but have failed. The cusped case is ok, since in that case there's a lower bound on the length of simple closed geodesics.

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The Heegaard genus of an arithmetic hyperbolic three-manifold is bounded below by a constant times its volume. This is proved on p43 of a recent paper by Gromov and Guth (or p2601 of the published version). So the answer to your question is yes, there are only finitely many arithmetic hyperbolic three-manifolds of any given Heegaard genus.

EDIT: as pointed out by Ian Agol, the definition of "arithmetic hyperbolic three-manifold" from the Gromov--Guth paper may be non-standard.

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This follows also from some results of Lackenby and lower bounds for first Laplace eigenvalues of arithmetic 3-manifolds (see Theorem 4.1 in people.maths.ox.ac.uk/lackenby/vh050905.pdf, and use Cheeger's inequality to connect the isoperimetric constant and Laplace eigenvalue). –  Denis Chaperon de Lauzières Aug 12 at 7:47
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Their argument only applies to congruence arithmetic manifolds. –  Ian Agol Aug 20 at 18:29
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I find it peculiar that this answer remains accepted and un-withdrawan, despite being wrong. –  Igor Rivin Aug 20 at 19:15
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Yes, usually an arithmetic manifold is defined up to commensurability, so may include non congruence lattices. If Neil intended the meaning in Gromov-Guth, then he should probably clarify, since it is nonstandard. –  Ian Agol Aug 21 at 3:38
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@JohnPardon: Sorry about the confusion here. I updated my question to include the definition I had in mind, which is how Ian answered the question. –  Neil Hoffman Aug 21 at 10:27

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