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In this paper we study existence and lack thereof of closed embedded orientable co-dimension one totally geodesic submanifolds of minimal volume cusped orientable hyperbolic manifolds., 16 pages, 2 figure

Throughout the paper,M will always denote a complete, orientable finite volume hyperbolic 3-manifold with cusps. By abuse of notation we will denote by ∂M to be the boundary of the compact manifold obtained from M by truncating the cusps. Given such a manifold, the SL2(C) character variety of M , X(M), is a complex algebraic set associated to representations of π1(M)→ SL2(C) (see §4 for more details). Work of Thurston showed that any irreducible component of such a variety containing the character of a discrete faithful representation has complex dimension equal to the number of cusps of M . Such components are called canonical components and are denoted X0(M). Character varieties have been fundamental tools in studying the topology of M (we refer the reader to [23] for more), and canonical components carry a wealth of topological information aboutM , including containing subvarieties associated to Dehn fillings of M . WhenM has exactly one cusp, any canonical component is a complex curve. The aim of this paper is to study how some of the natural invariants of these complex curves correspond to the underlying manifold M . In particular, we concentrate on how the gonality of these curves behaves in families of Dehn fillings on 2-cusped hyperbolic manifolds. More precisely, we study families of 1-cusped 3-manifolds which are obtained by Dehn filling of a single cusp of a fixed 2-cusped hyperbolic 3-manifold, M . We write M(−, r) to denote the manifold obtained by r = p/q filling of the second cusp of M . To state our results we introduce the following notation. If X is a complex curve, we write γ(X) to denote the gonality of X, g(X) to be the (geometric) genus of X and d(X) to be the degree (of the specified embedding) of X. The gonality of a curve is the lowest degree of a map from that curve to C. Unlike genus, gonality is not a topological invariant of curves, but rather is intimately connected to the geometry of the curve. There are connections between gonality and genus, most notably the Brill-Noether theorem which gives an upper bound for gonality in terms of genus (see § 7) but in some sense these are orthogonal invariants. For example, all hyperelliptic curves all have gonality two, but can have arbitrarily high genus. Moreover, for g > 2, there are curves of genus g of different gonality. We refer the reader to § 3 for precise definitions. Our first theorem is the following.

We give a new proof of a result of Sullivan [Hyperbolic geometry and homeomorphisms, inGeometric topology(ed. J. C. Cantrell), pp. 543–555 (Academic Press, New York, 1979)] establishing that all finite volume hyperbolicn-manifolds have a finite cover admitting a spin structure. In addition, in all dimensions greater than or equal to 5, we give the first examples of finite-volume hyperbolicn-manifolds that do not admit a spin structure.

In this article, we prove that the commensurability class of a closed, orientable, hyperbolic 3-manifold is determined by the surface subgroups of its fundamental group. Moreover, we prove that there can be only finitely many closed, orientable, hyperbolic 3-manifolds that have the same set of surfaces.

We prove that any arithmetic hyperbolic $n$-manifold of simplest type can either be geodesically embedded into an arithmetic hyperbolic $(n+1)$-manifold or its universal $\mathrm{mod}~2$ Abelian cover can., 20 pages; revised version, typos corrected; Mathematical Research Letters vol. 25, no. 4

We construct infinitely many examples of pairs of isospectral but non-isometric [Formula: see text]-cusped hyperbolic [Formula: see text]-manifolds. These examples have infinite discrete spectrum and the same Eisenstein series. Our constructions are based on an application of Sunada’s method in the cusped setting, and so in addition our pairs are finite covers of the same degree of a 1-cusped hyperbolic 3-orbifold (indeed manifold) and also have the same complex length spectra. Finally we prove that any finite volume hyperbolic 3-manifold isospectral to the figure-eight knot complement is homeomorphic to the figure-eight knot complement.

We prove that certain Fuchsian triangle groups are profinitely rigid in the absolute sense, i.e. each is distinguished from all other finitely generated, residually finite groups by its set of finite quotients. We also develop a method based on character varieties that can be used to distinguish between the profinite completions of certain groups., Comment: v1, 11 pages

We enumerate all the principal congruence link complements in $S^3$, there by answering a question of W. Thurston. Related articles: "Technical Report: All Principal Congruence Link Groups" (arXiv:1902.04722), "All Known Principal Congruence Links" (arXiv:1902.04426)., Comment: 7 pages, 1 table. V2: Addressing referee's comments, this article is now in announcement form, with technical details and link figures split off into separate articles

In this note, we show that there exist cusped hyperbolic 3-manifolds that embed geodesically, but cannot bound geometrically. Thus, being a geometric boundary is a non-trivial property for such manifolds. Our result complements the work by Long and Reid on geometric boundaries of compact hyperbolic 4-manifolds, and by Kolpakov, Reid and Slavich on embedding arithmetic hyperbolic manifolds., Nous montrons que plusieurs variétés hyperboliques en dimension trois aux bouts cuspides ne sont pas des bords géométriques. Alors, cette propriété est en fait suffisamment rare. Nos résultats augmentent le travail par Long et Reid dans le cas de variétés hyperboliques compactes en dimension trois qui fournissent des bords géodésiques pour les variétés hyperboliques compactes en dimensions quatre, et aussi le travail par Kolpakov, Reid et Slavich sur plongements géodésiques pour les variétés hyperboliques arithmétiques.

We construct arithmetic Kleinian groups that are profinitely rigid in the absolute sense: each is distinguished from all other finitely generated, residually finite groups by its set of finite quotients. The Bianchi group $\mathrm{PSL}(2,\mathbb{Z}[\omega])$ with $\omega^2+\omega+1=0$ is rigid in this sense. Other examples include the non-uniform lattice of minimal co-volume in $\mathrm{PSL}(2,\mathbb{C})$ and the fundamental group of the Weeks manifold (the closed hyperbolic $3$-manifold of minimal volume)., Comment: v2: 35 pages. Final version. To appear in the Annals of Mathematics, Vol. 192, no. 3, November 2020

International audience; In a previous article, we started an enumeration of the finitely many levels for which a principal congruence manifold can be a link complement in $S^3$. In this article we give a complete enumeration of all the principal congruence link complements in $S^3$, together with their levels in the case when the class number $Q(\sqrt{-d})$ of is greater than 1.

This paper describes in basic terms what a "Thin Group" is, as well as its uses in various subjects., Comment: 6 pages, 2 pictures

Let $M$ be a compact 3-manifold and $\Gamma=\pi_1(M)$. Work of Thurston and Culler--Shalen established the $\mathrm{SL}_2(\mathbb{C})$ character variety $X(\Gamma)$ as fundamental tool in the study of the geometry and topology of $M$. This is particularly the case when $M$ is the exterior of a hyperbolic knot $K$ in $S^3$. The main goals of this paper are to bring to bear tools from algebraic and arithmetic geometry to understand algebraic and number theoretic properties of the so-called canonical component of $X(\Gamma)$, as well as distinguished points on the canonical component, when $\Gamma$ is a knot group. In particular, we study how the theory of quaternion Azumaya algebras can be used to obtain algebraic and arithmetic information about Dehn surgeries, and perhaps of most interest, to construct new knot invariants that lie in the Brauer groups of curves over number fields., Comment: v2 Revisions and corrections; v3 To appear in IMRN

International audience; Cet article est consacré à un début d’énumération des compléments d’entrelacs dans S 3 provenant des groupes de congruence principaux. Nous utilisons des méthodes théoriques ainsi que des calculs avec MAGMA.

This paper is based on three lectures given by the author at the workshop, “Hyperbolic geometry and arithmetic: a crossview” held at The Universite Paul Sabatier, Toulouse in November 2012. The goal of the lectures was to describe recent work on the extent to which various geometric and analytical properties of hyperbolic 3-manifolds determine the commensurability class of such manifolds. This is the theme of the paper, and is for the most part a survey.

In this article we study the spectrum of totally geodesic surfaces of a finite volume hyperbolic 3-manifold. We show that for arithmetic hyperbolic 3-manifolds that contain a totally geodesic surface, this spectrum determines the commensurability class. In addition, we show that any finite volume hyperbolic 3-manifold has many pairs of non-isometric finite covers with identical spectra. Forgetting multiplicities, we can also construct pairs where the volume ratio is unbounded.

In this paper we provide a negative answer to a question of Farb about the relation between the algebraic degree of the stretch factor of a pseudo-Anosov homeomorphism and the genus of the surface on which it is defined., Comment: v2: Corrected minor errors

If $M$ is a compact 3-manifold whose first betti number is 1, and $N$ is a compact 3-manifold such that $\pi_1N$ and $\pi_1M$ have the same finite quotients, then $M$ fibres over the circle if and only if $N$ does. We prove that groups of the form $F_2\rtimes\mathbb{Z}$ are distinguished from one another by their profinite completions. Thus, regardless of betti number, if $M$ and $N$ are punctured torus bundles over the circle and $M$ is not homeomorphic to $N$, then there is a finite group $G$ such that one of $\pi_1M$ and $\pi_1N$ maps onto $G$ and the other does not., Comment: 17 pages, no figures. v2 minor corrections. This is the final version accepted for publication

Two groups are said to have the same nilpotent genus if they have the same nilpotent quotients. We answer four questions of Baumslag concerning nilpotent completions. (i) There exists a pair of finitely generated, residually torsion-free-nilpotent groups of the same nilpotent genus such that one is finitely presented and the other is not. (ii) There exists a pair of finitely presented, residually torsion-free-nilpotent groups of the same nilpotent genus such that one has a solvable conjugacy problem and the other does not. (iii) There exists a pair of finitely generated, residually torsion-free-nilpotent groups of the same nilpotent genus such that one has finitely generated second homology $H_2(-,\Z)$ and the other does not. (iv) A non-trivial normal subgroup of infinite index in a finitely generated parafree group cannot be finitely generated. In proving this last result, we establish that the first $L^2$ betti number of a finitely generated parafree group of rank $r$ is $r-1$. It follows that the reduced $C^*$-algebra of the group is simple if $r\ge 2$, and that a version of the Freiheitssatz holds for parafree groups., Comment: Version accepted for Int Math Res Notices

When G is a semi-simple Lie group, and Γ a lattice, a fundamental dichotomy established by Margulis [25], determines that CG(Γ) is dense in G if and only if Γ is arithmetic, and moreover, when Γ is non-arithmetic, CG(Γ) is again a lattice. Historically, the prominence of the commensurator was due in large part to the density of the commensurator in the arithmetic setting being closely related to the abundance of Hecke operators attached to arithmetic lattices. These operators are fundamental objects in the theory of automorphic forms associated to arithmetic lattices (see [38] for example). More recently, the commensurator of various classes of groups has come to the fore due to its growing role in geometry, topology and geometric group theory; for example in classifying lattices up to quasi-isometry, classifying graph manifolds up to quasi-isometry, and understanding Riemannian metrics admitting many “hidden symmetries” (for more on these and other topics see [2], [4], [17], [18], [24], [34] and [37]). In this article, we will study CG(Γ) when G = PSL(2,C) and Γ a finitely generated nonelementary Kleinian group. In this setting, we will abbreviate the notation for the commensurator of Γ to C(Γ). In the case that Γ is of finite co-volume and non-arithmetic, identifying C(Γ) has attracted considerable attention (see for example [19], [32] and [33]). Our focus here are those Kleinian groups Γ for which H/Γ has infinite volume. Henceforth, unless otherwise stated, the Kleinian groups Γ that we consider will always have infinite co-volume. If Γ is a Kleinian group, we denote by ΛΓ and ΩΓ the limit set and domain of discontinuity of Γ. The Kleinian group Γ is said to be of the first kind (resp. second kind) if ΩΓ = ∅ (resp. ΩΓ 6= ∅). The only known result for Kleinian groups Γ as above is due to L. Greenberg who proved the following result (see [20] and [21], and see §3.1 for a new proof when there are no parabolics).

The following problem that was posed by Grothendieck: Let uW H ! G be a homomorphism of nitely presented residually nite groups for which the extension O uW y H ! y G is an isomorphism. Is u an isomorphism? The problem was solved in the negative by Bridson and Grunewald who produced many examples of groups G and proper subgroups uW H, ! G for which O u is an isomorphism, but u is not. This paper addresses Grothendieck's problem in the context of 3-manifold groups.

We show that there exists a universal constant Kc so that every K-strongly quasiconformally homogeneous hyperbolic surface X (not equal to H2) has the property that K > Kc > 1. The constant Kc is the best possible, and is computed in terms of the diameter of the (2, 3, 7)-hyperbolic orbifold (which is the hyperbolic orbifold of smallest area.) We further show that the minimum strong homogeneity constant of a hyperbolic surface without conformal automorphisms decreases if one passes to a non-amenable regular cover.

A well-known conjecture about closed hyperbolic 3-manifolds asserts that the first Betti number can be increased without bound by passage to finite sheeted covers. If the manifold is fibred, a strengthening of this conjecture is that the number of fibred faces (see §2.1 for the definition of a fibred face) of the unit ball of the Thurston norm can be made arbitrarily large by passage to finite sheeted covers. The main result of this note is the following.

Let $\C(\Gamma)$ be the set of isomorphism classes of the finite groups that are homomorphic images of $\Gamma$. We investigate the extent to which $\C(\Gamma)$ determines $\Gamma$ when $\Gamma$ is a group of geometric interest. If $\Gamma_1$ is a lattice in ${\rm{PSL}}(2,\R)$ and $\Gamma_2$ is a lattice in any connected Lie group, then $\C(\Gamma_1) = \C(\Gamma_2)$ implies that $\Gamma_1$ is isomorphic to $\Gamma_2$. If $F$ is a free group and $\Gamma$ is a right-angled Artin group or a residually free group (with one extra condition), then $\C(F)=\C(\Gamma)$ implies that $F\cong\Gamma$. If $\Gamma_1, Comment: Minor edits. Version accepted by Israel J Math

We show that any finitely generated non-elementary Kleinian group has a co-final family of finite index normal subgroups with respect to which it has Property τ . As a consequence, any closed hyperbolic 3-manifold has a co-final family of finite index normal subgroups for which the infimal Heegaard gradient is positive.

This paper is based on a series of 4 lectures delivered at Groups St Andrews 2013. The main theme of the lectures was distinguishing finitely generated residually finite groups by their finite quotients. The purpose of this paper is to expand and develop the lectures. The paper is organized as follows. In §2 we collect some questions that motivated the lectures and this article, and in §3 discuss some examples related to these questions. In §4 we recall profinite groups, profinite completions and the formulation of the questions in the language of the profinite completion. In §5, we recall a particular case of the question of when groups have the same profinite completion, namely Grothendieck’s question. In §6 we discuss how the methods of L 2 -cohomology can be brought to bear on the questions in §2, and in §7, we give a similar discussion using the methods of the cohomology of profinite groups. In §8 we discuss the questions in §2 in the context of groups arising naturally in low-dimensional topology and geometry, and in §9 discuss parafree groups. Finally in §10 we collect a list of open problems that may be of interest.

We establish results concerning the profinite completions of 3-manifold groups. In particular, we prove that the complement of the figure-eight knot $S^3-K$ is distinguished from all other compact 3-manifolds by the set of finite quotients of its fundamental group. In addition, we show that if $M$ is a compact 3-manifold with $b_1(M)=1$, and $\pi_1(M)$ has the same finite quotients as a free-by-cyclic group $F_r\rtimes\mathbb{Z}$, then $M$ has non-empty boundary, fibres over the circle with compact fibre, and $\pi_1(M)\cong F_r\rtimes_\psi\mathbb{Z}$ for some $\psi\in{\rm{Out}}(F_r)$., Comment: 15 pages, no figures

If i is a flnite co-area Fuchsian group acting on H 2 , then the quotient H 2 =i is a hyperbolic 2-orbifold, with underlying space an orientable surface (possibly with punctures) and a flnite number of cone points. Through their close connections with number theory and the theory of automorphic forms, arithmetic Fuchsian groups form a widely studied and interesting subclass of flnite co-area Fuchsian groups. This paper is concerned with the distribution of arithmetic Fuchsian groups i for which the underlying surface of the orbifold H 2 =i is of genus zero; for short we say i is of genus zero. The motivation for the study of these groups comes from many difierent view

We show that for every n 2 there exists closed hyperbolic n-manifolds for which the full group of orientation preserving isometries is trivial. 2000 Mathematics Subject Classication 57S25, 57N16

We prove that any Coxeter group that is not virtually free contains a surface group. In particular if the Coxeter group is word hyperbolic and not virtually free this establishes the existence of a hyperbolic surface group, and answers in the affirmative a question of Gromov in this setting. We also discuss when Artin groups contain hyperbolic surface groups.

In this paper, we prove that if Γ \Gamma is a non-elementary subgroup of O o ( n , 1 , R ) \mathrm {O}_{\mathrm {o}}(n,1,\mathbb {R}) , with n ≥ 2 n\ge 2 , then the eigenvalue field of Γ \Gamma has infinite degree over Q \mathbb {Q} .

Let Od denote the ring of integers in [Formula: see text]. An orientable finite volume cusped hyperbolic 3-manifold M is called arithmetic if the faithful discrete representation of π1(M) into PSL(2,C) is conjugate to a group commensurable with some Bianchi group PSL (2,Od). If M is a closed orientable 3-manifold, we say a link L ⊂ M is arithmetic if M\L is arithmetic. In the paper, we show that there exist closed orientable 3-manifolds which do not contain anarithmetic knot. Our methods give much more precise informations for non-hyperbolic 3-manifolds. For certain Lens Spaces, we can give fairly complete statements.

We address the question that if π1-surjective maps between closed aspherical 3-manifolds have the same rank on π1 they must be of non-zero degree. The positive answer is proved for Seifert manifolds, which is used in constructing the first known example of minimal Haken manifold. Another motivation is to study epimorphisms of 3-manifold groups via maps of non-zero degree between 3-manifolds. Many examples are given.

Let H denote hyperbolic n-space, that is the unique connected simply connected Riemannian manifold of constant curvature −1. By a hyperbolic n-orbifold we shall mean a quotient H/Γ where Γ is a discrete group of isometries of H. If a hyperbolic n-manifold M is the totally geodesic boundary of a hyperbolic (n+1)-manifold W , we will say that M bounds geometrically. It was shown in [11] that if a closed orientable hyperbolic M4k−1 bounds geometrically, then η(M4k−1) ∈ Z. Closed hyperbolic 3-manifolds with integral eta are fairly rare – for example, of the 11, 000 or so manifolds in the census of small volume closed hyperbolic 3-manifolds, computations involving Snap (see [3]) rule out all but 41. (We refer the reader to [24] which contains the list of manifolds in the census with Chern-Simons invariant zero, as well as which of these have integral eta.) Hyperbolic 3-manifolds with totally geodesic boundary are fairly easily constructed given the Hyperbolization Theorem of Thurston [20], but to the authors’ knowledge, there was only one known prior example of a closed hyperbolic n-manifold (with n ≥ 3) which bounded geometrically, a somewhat ad hoc construction which appears in [18], based on a hyperbolic 4-manifold example due to Davis [4]. The difficulty is that almost nothing is known about hyperbolic manifolds in dimensions ≥ 4; some constructions exist (see [5], [6], [7]) but they do not appear to be sufficient to address this problem. This paper ameliorates this situation somewhat by providing a construction of examples in all dimensions. We show

We prove that certain hyperbolic Coxeter groups are separable on their geometrically finite subgroups.

D. Gabai, R. Meyerhoff and N. Thurston identified seven families of exceptional hyperbolic manifolds in their proof that a manifold which is homotopy equivalent to a hyperbolic manifold is hyperbolic. These families are each conjectured to consist of a single manifold. In fact, an important point in their argument depends on this conjecture holding for one particular exceptional family. In this paper, we prove the conjecture for that particular family, showing that the manifold known as Vol3 in the literature covers no other manifold. We also indicate techniques likely to prove this conjecture for five of the other six families.

Let M be a complete hyperbolic n-manifold of finite volume. By a systole of M we mean a shortest closed geodesic in M. By the systole length of M we mean the length of a systole. We denote this by sl (M). In the case when M is closed, the systole length is simply twice the injectivity radius of M. In the presence of cusps, injectivity radius becomes arbitrarily small and it is for this reason we use the language of ‘systole length’.In the context of hyperbolic surfaces of finite volume, much work has been done on systoles; we refer the reader to [2, 10–12] for some results. In dimension 3, little seems known about systoles. The main result in this paper is the following (see below for definitions)