Flexibility of surface groups in classical simple Lie groups

Inkang Kim; Pierre Pansu

Displaying similar documents to “Flexibility of surface groups in classical simple Lie groups”

Chen–Ruan Cohomology of $ℳ_{1, n}$ and ${\bar{ℳ}}_{1, n}$

Nicola Pagani (2013)

Annales de l’institut Fourier

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In this work we compute the Chen–Ruan cohomology of the moduli spaces of smooth and stable $n$ -pointed curves of genus $1$ . In the first part of the paper we study and describe stack theoretically the twisted sectors of $ℳ_{1, n}$ and ${\bar{ℳ}}_{1, n}$ . In the second part, we study the orbifold intersection theory of ${\bar{ℳ}}_{1, n}$ . We suggest a definition for an orbifold tautological ring in genus $1$ , which is a subring of both the Chen–Ruan cohomology and of the stringy Chow ring.

On surfaces with p $_{𝑔}$ = q = 1 and non-ruled bicanonical involution

Carlos Rito (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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This paper classifies surfaces $S$ of general type with $p_{g} = q = 1$ having an involution $i$ such that $S / i$ has non-negative Kodaira dimension and that the bicanonical map of $S$ factors through the double cover induced by $i .$ It is shown that $S / i$ is regular and either: a) the Albanese fibration of $S$ is of genus 2 or b) $S$ has no genus 2 fibration and $S / i$ is birational to a $K 3$ surface. For case a) a list of possibilities and examples are given. An example for case b) with $K^{2} = 6$ is also constructed.

$1$ -cocycles on the group of contactomorphisms on the supercircle $S^{1 | 3}$ generalizing the Schwarzian derivative

Boujemaa Agrebaoui, Raja Hattab (2016)

Czechoslovak Mathematical Journal

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The relative cohomology $H_{diff}^{1} (𝕂 (1 | 3), 𝔬𝔰𝔭 (2, 3); 𝒟_{λ, μ} (S^{1 | 3}))$ of the contact Lie superalgebra $𝕂 (1 | 3)$ with coefficients in the space of differential operators $𝒟_{λ, μ} (S^{1 | 3})$ acting on tensor densities on $S^{1 | 3}$ , is calculated in N. Ben Fraj, I. Laraied, S. Omri (2013) and the generating $1$ -cocycles are expressed in terms of the infinitesimal super-Schwarzian derivative $1$ -cocycle $s (X_{f}) = D_{1} D_{2} D_{3} (f) α_{3}^{1 / 2}$ , $X_{f} \in 𝕂 (1 | 3)$ which is invariant with respect to the conformal subsuperalgebra $𝔬𝔰𝔭 (2, 3)$ of $𝕂 (1 | 3)$ . In this work we study the supergroup case. We give an explicit construction of $1$ -cocycles...

Multiple end solutions to the Allen-Cahn equation in $ℝ^{2}$

Michał Kowalczyk, Yong Liu, Frank Pacard (2013-2014)

Séminaire Laurent Schwartz — EDP et applications

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An entire solution of the Allen-Cahn equation $Δ u = f (u)$ , where $f$ is an odd function and has exactly three zeros at $\pm 1$ and $0$ , e.g. $f (u) = u (u^{2} - 1)$ , is called a $2 k$ end solution if its nodal set is asymptotic to $2 k$ half lines, and if along each of these half lines the function $u$ looks (up to a multiplication by $- 1$ ) like the one dimensional, odd, heteroclinic solution $H$ , of $H^{''} = f (H)$ . In this paper we present some recent advances in the theory of the multiple end solutions. We begin with the description of the moduli space...

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...

Cluster ensembles, quantization and the dilogarithm

Vladimir V. Fock, Alexander B. Goncharov (2009)

Annales scientifiques de l'École Normale Supérieure

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A cluster ensemble is a pair $(𝒳, 𝒜)$ of positive spaces (i.e. varieties equipped with positive atlases), coming with an action of a symmetry group $Γ$ . The space $𝒜$ is closely related to the spectrum of a cluster algebra [12]. The two spaces are related by a morphism $p : 𝒜 \to 𝒳$ . The space $𝒜$ is equipped with a closed $2$ -form, possibly degenerate, and the space $𝒳$ has a Poisson structure. The map $p$ is compatible with these structures. The dilogarithm together with its motivic and quantum avatars plays a central...

Noetherian loop spaces

Natàlia Castellana, Juan Crespo, Jérôme Scherer (2011)

Journal of the European Mathematical Society

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The class of loop spaces of which the mod $p$ cohomology is Noetherian is much larger than the class of $p$ -compact groups (for which the mod $p$ cohomology is required to be finite). It contains Eilenberg–Mac Lane spaces such as $ℂ P^{\infty}$ and 3-connected covers of compact Lie groups. We study the cohomology of the classifying space $B X$ of such an object and prove it is as small as expected, that is, comparable to that of $B ℂ P^{\infty}$ . We also show that $B$ X differs basically from the classifying space of a $p$ -compact...

Invariants, torsion indices and oriented cohomology of complete flags

Baptiste Calmès, Viktor Petrov, Kirill Zainoulline (2013)

Annales scientifiques de l'École Normale Supérieure

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Let $G$ be a split semisimple linear algebraic group over a field and let $T$ be a split maximal torus of $G$ . Let $𝗁$ be an oriented cohomology (algebraic cobordism, connective $K$ -theory, Chow groups, Grothendieck’s $K_{0}$ , etc.) with formal group law $F$ . We construct a ring from $F$ and the characters of $T$ , that we call a formal group ring, and we define a characteristic ring morphism $c$ from this formal group ring to $𝗁 (G / B)$ where $G / B$ is the variety of Borel subgroups of $G$ . Our main result says that when the...

Batalin-Vilkovisky algebra structures on Hochschild cohomology

Luc Menichi (2009)

Bulletin de la Société Mathématique de France

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Let $M$ be any compact simply-connected oriented $d$ -dimensional smooth manifold and let $𝔽$ be any field. We show that the Gerstenhaber algebra structure on the Hochschild cohomology on the singular cochains of $M$ , $H H^{*} (S^{*} (M), S^{*} (M))$ , extends to a Batalin-Vilkovisky algebra. Such Batalin-Vilkovisky algebra was conjectured to exist and is expected to be isomorphic to the Batalin-Vilkovisky algebra on the free loop space homology on $M$ , $H_{* + d} (L M)$ introduced by Chas and Sullivan. We also show that the negative cyclic...

Symmetry of minimizers with a level surface parallel to the boundary

Giulio Ciraolo, Rolando Magnanini, Shigeru Sakaguchi (2015)

Journal of the European Mathematical Society

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We consider the functional $ℐ_{Ω} (v) = \int_{Ω} [f (| D v |) - v] d x,$ where $Ω$ is a bounded domain and $f$ is a convex function. Under general assumptions on $f$ , Crasta [Cr1] has shown that if $ℐ_{Ω}$ admits a minimizer in $W_{0}^{1, 1} (Ω)$ depending only on the distance from the boundary of $Ω$ , then $Ω$ must be a ball. With some restrictions on $f$ , we prove that spherical symmetry can be obtained only by assuming that the minimizer has one level surface parallel to the boundary (i.e. it has only a level surface in common with the distance). We then discuss...

Obstruction sets and extensions of groups

Francesca Balestrieri (2016)

Acta Arithmetica

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Let X be a nice variety over a number field k. We characterise in pure “descent-type” terms some inequivalent obstruction sets refining the inclusion $X {(_{k})}^{é t, B r} \subset X {(_{k})}^{B r ₁}$ . In the first part, we apply ideas from the proof of $X {(_{k})}^{é t, B r} = X {(_{k})}^{_{k}}$ by Skorobogatov and Demarche to new cases, by proving a comparison theorem for obstruction sets. In the second part, we show that if $\subset \subset_{k}$ are such that $\subset E x t (,_{k})$ , then $X {(_{k})}^{} = X {(_{k})}^{}$ . This allows us to conclude, among other things, that $X {(_{k})}^{é t, B r} = X {(_{k})}^{_{k}}$ and $X {(_{k})}^{S o l, B r ₁} = X {(_{k})}^{S o l_{k}}$ .

Effective bounds for Faltings’s delta function

Jay Jorgenson, Jürg Kramer (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

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In his seminal paper on arithmetic surfaces Faltings introduced a new invariant associated to compact Riemann surfaces $X$ , nowadays called Faltings’s delta function and here denoted by $δ_{Fal} (X)$ . For a given compact Riemann surface $X$ of genus $g_{X} = g$ , the invariant $δ_{Fal} (X)$ is roughly given as minus the logarithm of the distance with respect to the Weil-Petersson metric of the point in the moduli space $ℳ_{g}$ of genus $g$ curves determined by $X$ to its boundary $\partial ℳ_{g}$ . In this paper we begin by revisiting a formula derived...

SCAP-subalgebras of Lie algebras

Sara Chehrazi, Ali Reza Salemkar (2016)

Czechoslovak Mathematical Journal

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A subalgebra $H$ of a finite dimensional Lie algebra $L$ is said to be a $SCAP$ -subalgebra if there is a chief series $0 = L_{0} \subset L_{1} \subset ... \subset L_{t} = L$ of $L$ such that for every $i = 1, 2, ..., t$ , we have $H + L_{i} = H + L_{i - 1}$ or $H \cap L_{i} = H \cap L_{i - 1}$ . This is analogous to the concept of $SCAP$ -subgroup, which has been studied by a number of authors. In this article, we investigate the connection between the structure of a Lie algebra and its $SCAP$ -subalgebras and give some sufficient conditions for a Lie algebra to be solvable or supersolvable.

Pointed $k$ -surfaces

Graham Smith (2006)

Bulletin de la Société Mathématique de France

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Let $S$ be a Riemann surface. Let $ℍ^{3}$ be the $3$ -dimensional hyperbolic space and let $\partial_{\infty} ℍ^{3}$ be its ideal boundary. In our context, a Plateau problem is a locally holomorphic mapping $ϕ : S \to \partial_{\infty} ℍ^{3} = \hat{ℂ}$ . If $i : S \to ℍ^{3}$ is a convex immersion, and if $N$ is its exterior normal vector field, we define the Gauss lifting, $\hat{ı}$ , of $i$ by $\hat{ı} = N$ . Let $\vec{n} : U ℍ^{3} \to \partial_{\infty} ℍ^{3}$ be the Gauss-Minkowski mapping. A solution to the Plateau problem $(S, ϕ)$ is a convex immersion $i$ of constant Gaussian curvature equal to $k \in (0, 1)$ such that the Gauss lifting $(S, \hat{ı})$ is complete and $\vec{n} \circ \hat{ı} = ϕ$ . In this...

Fields of moduli of three-point $G$ -covers with cyclic $p$ -Sylow, II

Andrew Obus (2013)

Journal de Théorie des Nombres de Bordeaux

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We continue the examination of the stable reduction and fields of moduli of $G$ -Galois covers of the projective line over a complete discrete valuation field of mixed characteristic $(0, p)$ , where $G$ has a $p$ -Sylow subgroup $P$ of order $p^{n}$ . Suppose further that the normalizer of $P$ acts on $P$ via an involution. Under mild assumptions, if $f : Y \to ℙ^{1}$ is a three-point $G$ -Galois cover defined over $\bar{ℚ}$ , then the $n$ th higher ramification groups above $p$ for the upper numbering of the (Galois closure of...

Expansion in $S L_{d} (𝒪_{K} / I)$ , $I$ square-free

Péter P. Varjú (2012)

Journal of the European Mathematical Society

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Let $S$ be a fixed symmetric finite subset of $S L_{d} (𝒪_{K})$ that generates a Zariski dense subgroup of $S L_{d} (𝒪_{K})$ when we consider it as an algebraic group over $m a t h b b Q$ by restriction of scalars. We prove that the Cayley graphs of $S L_{d} (𝒪_{K} / I)$ with respect to the projections of $S$ is an expander family if $I$ ranges over square-free ideals of $𝒪_{K}$ if $d = 2$ and $K$ is an arbitrary numberfield, or if $d = 3$ and $K = ℚ$ .

A class of non-rational surface singularities with bijective Nash map

Camille Plénat, Patrick Popescu-Pampu (2006)

Bulletin de la Société Mathématique de France

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Let $(𝒮, 0)$ be a germ of complex analytic normal surface. On its minimal resolution, we consider the reduced exceptional divisor $E$ and its irreducible components $E_{i}$ , $i \in I$ . The Nash map associates to each irreducible component $C_{k}$ of the space of arcs through $0$ on $𝒮$ the unique component of $E$ cut by the strict transform of the generic arc in $C_{k}$ . Nash proved its injectivity and asked if it was bijective. As a particular case of our main theorem, we prove that this is the case if $E \cdot E_{i} < 0$ for any $i \in I$ . ...