Hyperplane sections in arithmetic hyperbolic manifolds

Bergeron, Nicolas; Haglund, Frédéric; Wise, Daniel T.

doi:10.1112/jlms/jdq082

Cited by 84 publications

(110 citation statements)

References 21 publications

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“…Combining the work of Bergeron-Haglund-Wise [3] and Bergeron-Wise [4], we have that there are virtually special closed hyperbolic manifolds in all dimensions.…”

Section: Theorem 1: Every Right-angled Artin Group Embeds Into Diffmentioning

confidence: 87%

Right-angled Artin groups in the C ∞ diffeomorphism group of the real line

Baik¹,

Kim²,

Koberda³

2016

Isr. J. Math.

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“…Combining the work of Bergeron-Haglund-Wise [3] and Bergeron-Wise [4], we have that there are virtually special closed hyperbolic manifolds in all dimensions.…”

Section: Theorem 1: Every Right-angled Artin Group Embeds Into Diffmentioning

confidence: 87%

Right-angled Artin groups in the C ∞ diffeomorphism group of the real line

Baik¹,

Kim²,

Koberda³

2016

Isr. J. Math.

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“…But it is conjectured that the fundamental group of every closed hyperbolic 3-manifold is LERF. A piece of evidence for this conjecture is given by the following important theorem, which is an amalgamation of work by Agol, Long and Reid [4] and Bergeron, Haglund and Wise [5]. There is an important new concept, introduced by Haglund and Wise [24], that relates to subgroup separability.…”

Section: Subgroup Separability Special Cube Complexes and Virtual Fimentioning

confidence: 98%

Finite Covering Spaces of 3-manifolds

Lackenby

2011

Proceedings of the International Congress of Mathematicians 2010 (ICM 2010)

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Abstract. Following Perelman's solution to the Geometrisation Conjecture, a 'generic' closed 3-manifold is known to admit a hyperbolic structure. However, our understanding of closed hyperbolic 3-manifolds is far from complete. In particular, the notorious Virtually Haken Conjecture remains unresolved. This proposes that every closed hyperbolic 3-manifold has a finite cover that contains a closed embedded orientable π1-injective surface with positive genus.I will give a survey on the progress towards this conjecture and its variants. Along the way, I will address other interesting questions, including: What are the main types of finite covering space of a hyperbolic 3-manifold? How many are there, as a function of the covering degree? What geometric, topological and algebraic properties do they have? I will show how an understanding of various geometric and topological invariants (such as the first eigenvalue of the Laplacian, the rank of mod p homology and the Heegaard genus) can be used to deduce the existence of π1-injective surfaces, and more.

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“…By applying LERFness of hyperbolic 3‐manifold (orbifold) groups, we get torsion free finite index subgroups

{normalΛ}_{i} < S O false(f_{i}; O_{K} false)

for

i = 1, 2

, with

{normalΛ}_{1} \cap S O_{0} false(f_{3}; O_{K} false) = {normalΛ}_{2} \cap S O_{0} false(f_{3}; O_{K} false) = Λ

, and

S = P / Λ

has large enough product neighborhood in

N_{1} = P_{1} / {normalΛ}_{1}

and

N_{2} = P_{2} / {normalΛ}_{2}

. Then there is an obvious map from

N_{1} \cup_{S} N_{2}

{double-struckH}^{4} / S O_{0} false(f; O_{K} false)

, and the induced map on the fundamental group

{normalΛ}_{1} *_{normalΛ} {normalΛ}_{2} \to S O_{0} false(f; O_{K} false)

is injective (for example, see [, Lemma 7.1]. So

S O_{0} false(f; O_{K} false)

contains a subgroup isomorphic to

{normalΛ}_{1} *_{normalΛ} {normalΛ}_{2}

.…”

Section: Nonlerfness Of Closed Arithmetic Hyperbolic 4‐manifold Groupsmentioning

confidence: 99%

“…By the same construction, we get a finite cover p 2 : N 2 → M 2 satisfying same properties. Actually, since we do not need condition (5) for N 2 , a simpler construction works.…”

Section: A Further Subgroup Of Algebraically Fiberedmentioning

confidence: 99%

Geometrically finite amalgamations of hyperbolic 3‐manifold groups are not LERF

Sun

2018

Proc. London Math. Soc.

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We prove that, for any two finite volume hyperbolic 3‐manifolds, the amalgamation of their fundamental groups along any nontrivial infinite index geometrically finite subgroup is not LERF (locally extended residually finite). This generalizes the author's previous work on nonLERFness of amalgamations of hyperbolic 3‐manifold groups along abelian subgroups. A consequence of this result is that, for all closed arithmetic hyperbolic 4‐manifolds have nonLERF fundamental groups. Along with the author's previous work, we obtain that, for any arithmetic hyperbolic manifold with dimension at least 4, with possible exceptions in seven‐dimensional manifolds defined by the octonion, its fundamental group is not LERF.

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Hyperplane sections in arithmetic hyperbolic manifolds

Cited by 84 publications

References 21 publications

Right-angled Artin groups in the C ∞ diffeomorphism group of the real line

Right-angled Artin groups in the C ∞ diffeomorphism group of the real line

Finite Covering Spaces of 3-manifolds

Geometrically finite amalgamations of hyperbolic 3‐manifold groups are not LERF

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