Elementary embeddings in torsion-free hyperbolic groups

Chloé Perin

Displaying similar documents to “Elementary embeddings in torsion-free hyperbolic groups”

Limits of relatively hyperbolic groups and Lyndon’s completions

Olga Kharlampovich, Alexei Myasnikov (2012)

Journal of the European Mathematical Society

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We describe finitely generated groups $H$ universally equivalent (with constants from $G$ in the language) to a given torsion-free relatively hyperbolic group $G$ with free abelian parabolics. It turns out that, as in the free group case, the group $H$ embeds into the Lyndon’s completion $G^{ℤ [t]}$ of the group $G$ , or, equivalently, $H$ embeds into a group obtained from $G$ by finitely many extensions of centralizers. Conversely, every subgroup of $G^{ℤ [t]}$ containing $G$ is universally equivalent to $G$ . Since finitely...

Quasi-isometric maps and Floyd boundaries of relatively hyperbolic groups

Victor Gerasimov, Leonid Potyagailo (2013)

Journal of the European Mathematical Society

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We describe the kernel of the canonical map from the Floyd boundary of a relatively hyperbolic group to its Bowditch boundary. Using the Floyd completion we further prove that the property of relative hyperbolicity is invariant under quasi-isometric maps. If a finitely generated group $H$ admits a quasi-isometric map $ϕ$ into a relatively hyperbolic group $G$ then $H$ is itself relatively hyperbolic with respect to a system of subgroups whose image under $ϕ$ is situated within a uniformly bounded...

Systole growth for finite area hyperbolic surfaces

Florent Balacheff, Eran Makover, Hugo Parlier (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

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In this note, we observe that the maximum value achieved by the systole function over all complete finite area hyperbolic surfaces of a given signature $(g, n)$ is greater than a function that grows logarithmically in terms of the ratio $g / n$ .

Hyperbolic spaces in Teichmüller spaces

Christopher J. Leininger, Saul Schleimer (2014)

Journal of the European Mathematical Society

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We prove, for any $n$ , that there is a closed connected orientable surface $S$ so that the hyperbolic space $ℍ^{n}$ almost-isometrically embeds into the Teichmüller space of $S$ , with quasi-convex image lying in the thick part. As a consequence, $ℍ^{n}$ quasi-isometrically embeds in the curve complex of $S$ .

Sur la classification des hexagones hyperboliques à angles droits en dimension 5

François Delgove, Nicolas Retailleau (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

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The aim of this paper is to give a classification of the right-angled hyperbolic hexagons in the real hyperbolic space $ℍ_{ℝ}^{5}$ , by using a quaternionic distance between geodesics in $ℍ_{ℝ}^{5}$ .

Fundamental solution $E_{n}$ of the operator $\partial^{2} / \partial t^{2} - Δ_{n}$ for n>3

Zofia Szmydt, Bogdan Ziemian (1979)

Annales Polonici Mathematici

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Large free subgroups of automorphism groups of ultrahomogeneous spaces

Szymon Głąb, Filip Strobin (2015)

Colloquium Mathematicae

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We consider the following notion of largeness for subgroups of $S_{\infty}$ . A group G is large if it contains a free subgroup on generators. We give a necessary condition for a countable structure A to have a large group Aut(A) of automorphisms. It turns out that any countable free subgroup of $S_{\infty}$ can be extended to a large free subgroup of $S_{\infty}$ , and, under Martin’s Axiom, any free subgroup of $S_{\infty}$ of cardinality less than can also be extended to a large free subgroup of $S_{\infty}$ . Finally, if Gₙ are countable...

Computing the determinantal representations of hyperbolic forms

Mao-Ting Chien, Hiroshi Nakazato (2016)

Czechoslovak Mathematical Journal

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The numerical range of an $n \times n$ matrix is determined by an $n$ degree hyperbolic ternary form. Helton-Vinnikov confirmed conversely that an $n$ degree hyperbolic ternary form admits a symmetric determinantal representation. We determine the types of Riemann theta functions appearing in the Helton-Vinnikov formula for the real symmetric determinantal representation of hyperbolic forms for the genus $g = 1$ . We reformulate the Fiedler-Helton-Vinnikov formulae for the genus $g = 0, 1$ , and present an elementary...

Hyperideal polyhedra in hyperbolic 3-space

Xiliang Bao, Francis Bonahon (2002)

Bulletin de la Société Mathématique de France

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A hyperideal polyhedron is a non-compact polyhedron in the hyperbolic $3$ -space $ℍ^{3}$ which, in the projective model for $ℍ^{3} \subset {ℝℙ}^{3}$ , is just the intersection of $ℍ^{3}$ with a projective polyhedron whose vertices are all outside $ℍ^{3}$ and whose edges all meet $ℍ^{3}$ . We classify hyperideal polyhedra, up to isometries of $ℍ^{3}$ , in terms of their combinatorial type and of their dihedral angles.

The existence of Carathéodory solutions of hyperbolic functional differential equations

Adrian Karpowicz (2010)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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We consider the following Darboux problem for the functional differential equation $\partial ² u / \partial x \partial y (x, y) = f (x, y, u_{(x, y)}, \partial u / \partial x (x, y), \partial u / \partial y (x, y))$ a.e. in [0,a]×[0,b], u(x,y) = ψ(x,y) on [-a₀,a]×[-b₀,b] $0, a] \times (0, b],$ where the function $u_{(x, y)} : [- a ₀, 0] \times [- b ₀, 0] \to ℝ^{k}$ is defined by $u_{(x, y)} (s, t) = u (s + x, t + y)$ for (s,t) ∈ [-a₀,0]×[-b₀,0]. We prove a theorem on existence of the Carathéodory solutions of the above problem.

On fixed point free involutions of $R^{1} \times S^{2}$

Gerhard X. Ritter (1976)

Colloquium Mathematicae

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An invariance method for constructing fundamental solutions for $P (□_{m} n)$

Zofia Szmydt, Bogdan Ziemian (1985)

Annales Polonici Mathematici

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On the structural theory of ${II}_{1}$ factors of negatively curved groups

Ionut Chifan, Thomas Sinclair (2013)

Annales scientifiques de l'École Normale Supérieure

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Ozawa showed in [21] that for any i.c.c. hyperbolic group, the associated group factor $L Γ$ is solid. Developing a new approach that combines some methods of Peterson [29], Ozawa and Popa [27, 28], and Ozawa [25], we strengthen this result by showing that $L Γ$ is strongly solid. Using our methods in cooperation with a cocycle superrigidity result of Ioana [12], we show that profinite actions of lattices in $Sp (n, 1)$ , $n \geq 2$ , are virtually $W^{*}$ -superrigid.

Hyperbolic geometry and moduli of real cubic surfaces

Daniel Allcock, James A. Carlson, Domingo Toledo (2010)

Annales scientifiques de l'École Normale Supérieure

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Let $ℳ_{0}^{ℝ}$ be the moduli space of smooth real cubic surfaces. We show that each of its components admits a real hyperbolic structure. More precisely, one can remove some lower-dimensional geodesic subspaces from a real hyperbolic space $H^{4}$ and form the quotient by an arithmetic group to obtain an orbifold isomorphic to a component of the moduli space. There are five components. For each we describe the corresponding lattices in $PO (4, 1)$ . We also derive several new and several old results on the topology...

A characterization of Fuchsian groups acting on complex hyperbolic spaces

Xi Fu, Liulan Li, Xiantao Wang (2012)

Czechoslovak Mathematical Journal

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Let $G \subset 𝐒𝐔 (2, 1)$ be a non-elementary complex hyperbolic Kleinian group. If $G$ preserves a complex line, then $G$ is $ℂ$ -Fuchsian; if $G$ preserves a Lagrangian plane, then $G$ is $ℝ$ -Fuchsian; $G$ is Fuchsian if $G$ is either $ℂ$ -Fuchsian or $ℝ$ -Fuchsian. In this paper, we prove that if the traces of all elements in $G$ are real, then $G$ is Fuchsian. This is an analogous result of Theorem V.G. 18 of B. Maskit, Kleinian Groups, Springer-Verlag, Berlin, 1988, in the setting of complex hyperbolic isometric groups. As an...

A multiparameter variant of the Salem-Zygmund central limit theorem on lacunary trigonometric series

Mordechay B. Levin (2013)

Colloquium Mathematicae

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We prove the central limit theorem for the multisequence $\sum_{1 \leq n ₁ \leq N ₁} \dots \sum_{1 \leq n_{d} \leq N_{d}} a_{n ₁, . . ., n_{d}} c o s (⟨ 2 π m, A ₁^{n ₁} . . . A_{d}^{n_{d}} x ⟩)$ where $m \in ℤ^{s}$ , $a_{n ₁, . . ., n_{d}}$ are reals, $A ₁, . . ., A_{d}$ are partially hyperbolic commuting s × s matrices, and x is a uniformly distributed random variable in ${[0, 1]}^{s}$ . The main tool is the S-unit theorem.

Prescribing endomorphism algebras of $ℵ_{n}$ -free modules

Rüdiger Göbel, Daniel Herden, Saharon Shelah (2014)

Journal of the European Mathematical Society

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It is a well-known fact that modules over a commutative ring in general cannot be classified, and it is also well-known that we have to impose severe restrictions on either the ring or on the class of modules to solve this problem. One of the restrictions on the modules comes from freeness assumptions which have been intensively studied in recent decades. Two interesting, distinct but typical examples are the papers by Blass [1] and Eklof [8], both jointly with Shelah. In the first case...

Bounded cohomology and isometry groups of hyperbolic spaces

Ursula Hamenstädt (2008)

Journal of the European Mathematical Society

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Let $X$ be an arbitrary hyperbolic geodesic metric space and let $Γ$ be a countable subgroup of the isometry group $Iso (X)$ of $X$ . We show that if $Γ$ is non-elementary and weakly acylindrical (this is a weak properness condition) then the second bounded cohomology groups $H_{b}^{2} (Γ, ℝ)$ , $H_{b}^{2} (Γ, ℓ^{p} (Γ))$ $(1 < p < \infty)$ are infinite dimensional. Our result holds for example for any subgroup of the mapping class group of a non-exceptional surface of finite type not containing a normal subgroup which virtually splits as a direct...