Geometrically infinite surfaces with discrete length spectra

It is well-known that the length spectrum of a geometrically finite hyperbolic manifold is discrete. In this paper, we begin a study of the length spectrum for geometrically infinite hyperbolic surfaces. In this generality, it is possible that the spectrum is not discrete and the main focus of this work is to find necessary and sufficient conditions for a geometrically infinite surface to have a discrete spectrum. After deriving a number of properties of the length spectrum, we show that every topological surface of infinite type admits both an infinite dimensional family of quasiconformally distinct hyperbolic structures having a discrete length spectrum, and an infinite dimensional family of quasiconformally distinct structures with a nondiscrete spectrum. Moreover, there exists such an infinite dimensional subspace arbitrarily close to (in the Fenchel-Nielsen topology) any hyperbolic structure.      

How many different cascades on a surface can have coinciding hyperbolic attractors?

It is shown that the number of essentially nonconjugate (i.e., not being iterations of topologically conjugate) diffeomorphisms of a surface having homeomorphic one-dimensional hyperbolic attractors can be arbitrarily large, provided that the genus of the surface is large enough. A lower bound for this number depending on the surface genus is given. The corresponding result for pseudo-Anosov homeomorphisms is stated.     

Finite-volume hyperbolic 4-manifolds that share a fundamental polyhedron

It is known that the volume function for hyperbolic manifolds of dimension 3 is finite-to-one. We show that the number of nonhomeomorphic hyperbolic 4-manifolds with the same volume can be made arbitrarily large. This is done by constructing a sequence of finite-sided finite-volume polyhedra with side-pairings that yield manifolds. In fact, we show that arbitrarily many nonhomeomorphic hyperbolic 4-manifolds may share a fundamental polyhedron. As a by-product of our examples, we also show in a constructive way that the set of volumes of hyperbolic 4-manifolds contains the set of even integral multiples of 4π²/3. This is “half” the set of possible values for volumes, which is the integral multiples of 4π²/3 due to the Gauss-Bonnet formula Vol(M) = 4π²/3 · χ(M).     

Embeddings of SL(2; ?) into the cremona group

Geometric and dynamic properties of embeddings of SL(2; ℤ) into the Cremona group are studied. Infinitely many nonconjugate embeddings that preserve the type (i.e., that send elliptic, parabolic and hyperbolic elements onto elements of the same type) are provided. The existence of infinitely many nonconjugate elliptic, parabolic and hyperbolic embeddings is also shown. In particular, a group G of automorphisms of a smooth surface S obtained by blowing up 10 points of the complex projective plane is given. The group G is isomorphic to SL(2; ℤ), preserves an elliptic curve and all its elements of infinite order are hyperbolic.     

Realizing Alexander polynomials by hyperbolic links

We realize a given (monic) Alexander polynomial by a (fibered) hyperbolic arborescent knot and link having any number of components, and by infinitely many such links having at least 4 components. As a consequence, a Mahler measure minimizing polynomial, if it exists, is realized as the Alexander polynomial of a fibered hyperbolic link of at least 2 components. For a given polynomial, we also give an upper bound for the minimal hyperbolic volume of knots/links realizing the polynomial and, in the opposite direction, construct knots of arbitrarily large volume, which are arborescent, or have given free genus at least 2.     

Commutator length of symplectomorphisms

Each element $x$ of the commutator subgroup $[G, G]$ of a group $G$ can be represented as a product of commutators. The minimal number of factors in such a product is called the commutator length of $x$. The commutator length of $G$ is defined as the supremum of commutator lengths of elements of $[G, G]$. We show that for certain closed symplectic manifolds $(M,\omega)$, including complex projective spaces and Grassmannians, the universal cover $\widetilde{\hbox{\rm Ham}\, (M,\omega)$ of the group of Hamiltonian symplectomorphisms of $(M,\omega)$ has infinite commutator length. In particular, we present explicit examples of elements in $\widetilde{\hbox{\rm Ham}\, (M,\omega)$ that have arbitrarily large commutator length -- the estimate on their commutator length depends on the multiplicative structure of the quantum cohomology of $(M,\omega)$. By a different method we also show that in the case $c_1 (M) = 0$ the group $\widetilde{\hbox{\rm Ham}\, (M,\omega)$ and the universal cover ${\widetilde{\Symp}}_0\, (M,\omega)$ of the identity component of the group of symplectomorphisms of $(M,\omega)$ have infinite commutator length.     

Hyperbolic Coxeter groups of large dimension

We construct examples of Gromov hyperbolic Coxeter groups of arbitrarily large dimension. We also extend Vinbergs theorem to show that if a Gromov hyperbolic Coxeter group is a virtual Poincaré duality group of dimension n, then n 61.Coxeter groups acting on their associated complexes have been extremely useful source of examples and insight into nonpositively curved spaces over last several years. Negatively curved (or Gromov hyperbolic) Coxeter groups were much more elusive. In particular their existence in high dimensions was in doubt.In 1987 Gabor Moussong [M] conjectured that there is a universal bound on the virtual cohomological dimension of any Gromov hyperbolic Coxeter group. This question was also raised by Misha Gromov [G] (who thought that perhaps any construction of high dimensional negatively curved spaces requires nontrivial number theory in the guise of arithmetic groups in an essential way), and by Mladen Bestvina [B2].In the present paper we show that high dimensional Gromov hyperbolic Coxeter groups do exist, and we construct them by geometric or group theoretic but not arithmetic means.     

Quasimorphisms,random walks,and transient subsets in countable groups

We study the interrelations between the theory of quasimorphisms and the theory of random walks on groups, and establish the following transience criterion for subsets of groups: if a subset of a countable group has bounded images under any three linearly independent homogeneous quasimorphisms on the group, the this subset is transient for all nondegenerate random walks on the group. From this it follows, by results of M. Bestvina, K. Fujiwara, J. Birman, W. Menasco, and others, that, in a certain sense, generic elements in the mapping class groups of surfaces are pseudo-Anosov, generic braids in Artin’s braid groups represent prime links and knots, generic elements in the commutant of every nonelementary hyperbolic group have large stable commutator length, etc. Bibliography: 20 titles.     

Isometry Groups of Hyperbolic Three-Manifolds which are Cyclic Branched Coverings

Let M be a hyperbolic three-manifold which is an n-fold cyclic branched covering of a hyperbolic link L in the three-sphere, or more precisely, of the hyperbolic three-orbifold whose underlying topological space is the three-sphere and whose singular set, of branching index n, the link L. We say that M has no hidden symmetries (with respect to the given branched covering) if the isometry group of M is the lift of (a subgroup of) the isometry group of the hyperbolic orbifold (which is isomorphic to the symmetry group of the link L). It follows from Thurston's hyperbolic surgery theorem that M has no hidden symmetries if n is sufficiently large. Our main result is an explicit numerical version of this fact: we give a constant, in terms of the volume of the complement of L, such that M has no hidden symmetries for all n larger than this constant; we show by examples that a universal constant working for all hyperbolic knots or links does not exist. We give also some results on the possible orders and the structure of the isometry group of M. Finally, we construct sets of four different -hyperbolic knots which have the same two-fold branched covering (a hyperbolic three-manifold); it is an interesting question for how many different -hyperbolic knots (or links) this may happen (in the case of hyperbolic knots, for arbitrarily many).     

On the distribution of Farey fractions and hyperbolic lattice points

We derive an asymptotic formula for the number of pairs of consecutive fractions a0=q0 and a=q in the Farey sequence of order Q such that a=q; q=Q; and (Q - q0)=q lie each in prescribed subintervals of the interval [0; 1]. We deduce the leading term in the asymptotic formula for `the hyperbolic lattice point problem" for the modular group PSL(2; Z ), the number of images of a given point under the action of the group in a given circle in the hyperbolic plane.     

Moduli spaces of local systems and higher Teichmüller theory

Let G be a split semisimple algebraic group over Q with trivial center. Let S be a compact oriented surface, with or without boundary. We define positive representations of the fundamental group of S to G(R), construct explicitly all positive representations, and prove that they are faithful, discrete, and positive hyperbolic; the moduli space of positive representations is a topologically trivial open domain in the space of all representations. When S have holes, we defined two moduli spaces closely related to the moduli spaces of G-local systems on S. We show that they carry a lot of interesting structures. In particular we define a distinguished collection of coordinate systems, equivariant under the action of the mapping class group of S. We prove that their transition functions are subtraction free. Thus we have positive structures on these moduli spaces. Therefore we can take their points with values in any positive semifield. Their positive real points provide the two higher Teichmüller spaces related to G and S, while the points with values in the tropical semifields provide the lamination spaces. We define the motivic avatar of the Weil–Petersson form for one of these spaces. It is related to the motivic dilogarithm.     

Word length in surface groups with characteristic generating sets

A subset of a group is characteristic if it is invariant under every automorphism of the group. We study word length in fundamental groups of closed hyperbolic surfaces with respect to characteristic generating sets consisting of a finite union of orbits of the automorphism group, and show that the translation length of any element with a nonzero crossing number is positive, and bounded below by a constant depending only (and explicitly) on a bound on the crossing numbers of generating elements. This answers a question of Benson Farb.

     

Relative hyperbolicity and Artin groups

This paper considers the question of relative hyperbolicity of an Artin group with regard to the geometry of its associated Deligne complex. We prove that an Artin group is weakly hyperbolic relative to its finite (or spherical) type parabolic subgroups if and only if its Deligne complex is a Gromov hyperbolic space. For a two-dimensional Artin group the Deligne complex is Gromov hyperbolic precisely when the corresponding Davis complex is Gromov hyperbolic, that is, precisely when the underlying Coxeter group is a hyperbolic group. For Artin groups of FC type we give a sufficient condition for hyperbolicity of the Deligne complex which applies to a large class of these groups for which the underlying Coxeter group is hyperbolic. The key tool in the proof is an extension of the Milnor-Svarc Lemma which states that if a group G admits a discontinuous, co-compact action by isometries on a Gromov hyperbolic metric space, then G is weakly hyperbolic relative to the isotropy subgroups of the action.      

A new family of representations of virtually free groups

We construct a new family of irreducible unitary representations of a finitely generated virtually free group Λ. We prove furthermore a general result concerning representations of Gromov hyperbolic groups that are weakly contained in the regular representation, thus implying that all the representations in this family can be realized on the boundary of Λ. As a corollary, we obtain an analogue of Herz majorization principle.     

图上Z~2可扩作用的非存在性

文[9]中作者考虑连续统上可扩群作用的存在性问题,证明了单位闭区间上不存在自由交换群Z×Z的可扩作用,并且给出一个例子表明闭区间上存在自由积Z*Z的可扩作用.换句话说,由两个交换同胚生成的群是不能可扩作用在闭区间上的,但还是存在由两个非交换同胚生成的群能够可扩作用在闭区间上.本文证明了图上不存在Z×Z的可扩作用,解决了文[9]所提的一个问题.     

Complex hyperbolic Kleinian groups with limit set a wild knot

We construct quasi-Fuchsian groups acting on two-dimensional complex hyperbolic space with limit set a wild knot. Also, we study the Teichmüller space T(G) of faithful, discrete, type-preserving representations of a Fuchsian group G of the first kind with parabolic elements in complex hyperbolic space. We show that T(G) is not connected, and that the Toledo invariant does not distinguish different connected components of T(G).     

Selmer groups of elliptic curves that can be arbitrarily large

In this article, it is shown that certain kinds of Selmer groups of elliptic curves can be arbitrarily large. The main result is that if p is a prime at least 5, then p-Selmer groups of elliptic curves can be arbitrarily large if one ranges over number fields of degree at most g+1 over the rationals, where g is the genus of X₀(p). As a corollary, one sees that p-Selmer groups of elliptic curves over the rationals can be arbitrarily large for p=5,7 and 13 (the cases p?7 were already known). It is also shown that the number of elements of order N in the N-Selmer group of an elliptic curve over the rationals can be arbitrarily large for N=9,10,12,16 and 25.     

Length and Stable Length

This paper establishes the existence of a gap for the stable length spectrum on a hyperbolic manifold. If M is a hyperbolic n-manifold, for every positive ϵ there is a positive δ depending only on n and on ϵ such that an element of π₁(M) with stable commutator length less than δ is represented by a geodesic with length less than ϵ. Moreover, for any such M, the first accumulation point for stable commutator length on conjugacy classes is at least 1/12. Conversely, “most” short geodesics in hyperbolic 3-manifolds have arbitrarily small stable commutator length. Thus stable commutator length is typically good at detecting the thick-thin decomposition of M, and 1/12 can be thought of as a kind of homological Margulis constant. Received: June 2006 Revision: May 2007 Accepted: June 2007     

Infinitely many meridional essential surfaces of bounded genus in hyperbolic knot exteriors

Geometriae Dedicata - We show the existence of an infinite collection of hyperbolic knots where each of which has in its exterior meridional essential planar surfaces of arbitrarily large number of...     

Minimal Lagrangian surfaces in {\mathbb {CH}^2}and representations of surface groups into SU(2, 1)

We use an elliptic differential equation of ?i?eica (or Toda) type to construct a minimal Lagrangian surface in ${\mathbb {CH}^2}$ from the data of a compact hyperbolic Riemann surface and a cubic holomorphic differential. The minimal Lagrangian surface is equivariant for an SU(2, 1) representation of the fundamental group. We use this data to construct a diffeomorphism between a neighbourhood of the zero section in a holomorphic vector bundle over Teichmuller space (whose fibres parameterise cubic holomorphic differentials) and a neighborhood of the ${\mathbb {R}}$ -Fuchsian representations in the SU(2, 1) representation space. We show that all the representations in this neighbourhood are complex-hyperbolic quasi-Fuchsian by constructing for each a fundamental domain using an SU(2, 1) frame for the minimal Lagrangian immersion: the Maurer–Cartan equation for this frame is the ?i?eica-type equation. A very similar equation to ours governs minimal surfaces in hyperbolic 3-space, and our paper can be interpreted as an analog of the theory of minimal surfaces in quasi-Fuchsian manifolds, as first studied by Uhlenbeck.