bluuebunny/crossref_metadata_2025 huggingface dataset:
Witt groups of sheaves on topological spaces
Authors: Woolf Jon
Published: December 2009

This paper investigates the Witt groups of triangulated categories of sheaves (of modules over a ring R in which 2 is invertible) equipped with Poincaré–Verdier duality. We consider two main cases, that of perfect complexes of sheaves on locally compact Hausdorff spaces and that of cohomologically constructible complexes of sheaves on polyhedra. We show that the Witt groups of the latter form a generalised homology theory for polyhedra and continuous maps. Under certain restrictions on the ring R, we identify these constructible Witt groups of a finite simplicial complex with Ranicki’s free symmetric L-groups. Witt spaces are the natural class of spaces for which the rational intersection homology groups have Poincaré duality. When the ring R is the rationals we identify the constructible Witt groups with the 4-periodic colimit of the bordism groups of PL Witt spaces. This allows us to interpret L-classes of singular spaces as stable homology operations from the constructible Witt groups to rational homology.
A volume maximizing canonical surface in 3-space
Authors: Bauer Ingrid , Catanese Fabrizio
Published: December 2009

Answering a question posed by Enriques, we construct a minimal smooth algebraic surface S of general type over the complex numbers with K 2 = 45 and pg = 4, and with birational canonical map. The canonical system |K S| has a fixed part and the degree of the canonical image is 19. The surface we construct is rigid, S is indeed a ball quotient. It is obtained as an Abelian covering of the plane branched over an arrangement of lines already considered by Hirzebruch, and it is the first such example which is regular (q = 0).
The Hopf boundary point lemma for vector bundle sections
Authors: Pulemotov Artem
Published: December 2009

The paper establishes a version of the Hopf boundary point lemma for sections of a vector bundle over a manifold with boundary. This result may be viewed as a counterpart to the tensor maximum principle obtained by R. Hamilton in 1986. Potential applications include the study of various geometric flows and the construction of invariant sets for geometric boundary value problems.
On the space of metrics with invertible Dirac operator
Authors: Dahl Mattias
Published: December 2009

On a compact spin manifold we study the space of Riemannian metrics for which the Dirac operator is invertible. The first main result is a surgery theorem stating that such a metric can be extended over the trace of a surgery of codimension at least three. We then prove that, if non-empty, the space of metrics with invertible Dirac operators is disconnected in dimensions

n ≡ 0,1,3,7

mod

8

,

n ≥ 5

. As corollaries follow results on the existence of metrics with harmonic spinors by Hitchin and Bär. Finally we use computations of the eta invariant by Botvinnik and Gilkey to find metrics with harmonic spinors on simply connected manifolds with a cyclic group action. In particular this applies to spheres of all dimensions

n ≥ 5

.
Prescription du spectre du laplacien de Hodge–de Rham dans une classe conforme
Authors: Jammes Pierre
Published: December 2009

For any compact manifold of dimension

n ≥ 5

, we prescribe the volume and any finite part of the spectrum of the Hodge Laplacian (without multiplicity) acting on differential forms of degree

p \in [2,n − 2]

(except for

p = n/2

if

n

is even), within a given conformal class. When

n ≤ 4

and when

p = 0, 1, n − 1, n

, and

p = n/2

if

n

is even, this simultaneous prescription of the volume, the spectrum and the conformal class is known to be impossible.

Sur toute variété compacte de dimension

n ≥ 5

, on prescrit le volume et toute partie finie du spectre du laplacien de Hodge–de Rham (sans multiplicité) en restriction aux formes de degré

p \in [2,n − 2]

, en excluant

p = n/2

si

n

est pair, et en imposant à la métrique d'appartenir à une classe conforme donnée. On sait que pour

n ≤ 4

, ainsi que pour

p = 0, 1, n − 1, n

, et

p = n/2

si

n

est pair, on ne peut pas prescrire simultanément le spectre, le volume et la classe conforme.
Rigidité topologique sous l'hypothèse « entropie majorée » et applications
Authors: Reviron Guillemette
Published: December 2009

We study some families of compact length spaces whose entropy is bounded from above. We prove that these families are complete w.r.t. the Gromov–Hausdorff distance and we give an explicit constant ε0 > 0 such that, on balls of radius ε0 with respect to the Gromov–Hausdorff distance, the fundamental group is constant, the universal covers are close for the equivariant Gromov–Hausdorff distance, the length spectrum is continuous and the entropy is Lipschitz. If we consider now some subsets of manifolds, we show moreover that the volume is semi-continuous from below and that the integral of the Ricci curvature is bounded from below.

Nous étudions certaines familles d'espaces de longueur compacts dont l'entropie volumique est majorée. Nous montrons que ces familles sont complètes pour la distance de Gromov–Hausdorff et nous prouvons l'existence d'une constante explicite ε0 > 0 telle que, sur les boules de rayon ε0 pour la distance de Gromov–Hausdorff, le groupe fondamental est constant, les revêtements universels sont proches pour la distance de Gromov–Hausdorff équivariante, le spectre des longueurs est continu, l'entropie est Lipschitzienne. Si l'on se restreint à certains sous-ensembles des variétés riemanniennes compactes, nous montrons de plus que, sur ces boules de rayon ε0, le volume est semi-continu inférieurement et que l'intégrale de la courbure de Ricci est minorée uniformément.
Irreducibly represented groups
Authors: Bekka Bachir , de la Harpe Pierre
Published: December 2009

A group is
irreducibly represented
if it has a faithful irreducible unitary representation. For countable groups, a criterion for irreducible representability is given, which generalises a result obtained for finite groups by W. Gaschütz in 1954. In particular, torsionfree groups and infinite conjugacy class groups are irreducibly represented.

We indicate some consequences of this for operator algebras. In particular, we characterise up to isomorphism the countable subgroups

Δ

of the unitary group of a separable infinite dimensional Hilbert space

\mathcal H

of which the bicommutants

Δ''

(in the sense of the theory of von Neumann algebras) coincide with the algebra of all bounded linear operators on

\mathcal H

.
Reparamétrisation universelle de familles f-analytiques de cycles et théorème de f-aplatissement géométrique
Authors: Barlet Daniel
Published: December 2009

This article presents a new point of view around the main results of D. Mathieu [M00] on meromorphic equivalence relations. We introduce the space of finite type cycles (closed analytic cycles with finitely many irreducible components) of a given finite dimensional complex space and a natural topology on this space, in order to avoid the “regularity” condition for analytic families of cycles introduced in loc. cit. and also the two notions of “escape to infinity” which are here encoded in a natural way in our framework. Then the results are stronger and much simpler to state and to use. They contain, in a slightly different language, a clean and more general version of the works of H. Grauert [G83] and [G86] and of B. Siebert [S93] and [S94] on meromorphic equivalence relations.

Cet article présente un nouveau point de vue à propos des principaux résultats de David Mathieu [M00] sur les relations d'équivalence méromorphes. Nous introduisons l'espace des cycles de type fini (les cycles analytiques fermés n'ayant qu'un nombre fini de composantes irréductibles) d'un espace analytique complexe donné de dimension finie, muni d'une topologie naturelle, ce qui permet d'éviter la condition de (« régularité » des familles analytiques de cycles qui est utilisée dans loc. cit. et également les deux notions de « fuite à l'infini » qui sont ici encodées de façon naturelle dans notre contexte. Les résultats obtenus sont meilleurs et surtout d'énoncés et d'utilisation beaucoup plus simples. Ils contiennent, avec un langage un peu différent, une version plus claire et plus générale des travaux de H. Grauert [G83] et [G86] et de B. Siebert [S93] et [S94] sur les relations d'équivalence méromorphes.
On the Kneser–Tits problem for triality forms
Authors: Prasad Gopal
Published: December 2009

The purpose of this paper is to provide a concrete description of the “Whitehead group” W(k,G) := G(k)/G(k)+ for the simply connected triality forms G of k-rank 1, and to use this description to prove that if k is a global field, then the Kneser–Tits problem for these forms has an affirmative solution.
The macroscopic spectrum of nilmanifolds with an emphasis on the Heisenberg groups
Authors: Vernicos Constantin
Published: December 2009

Take a Riemannian nilmanifold, lift its metric on its universal cover. In that way one obtains a metric invariant under the action of some co-compact subgroup. We use it to define metric balls and then study the spectrum of the Dirichlet Laplacian. Using homogenization techniques we describe the asymptotic behavior of the spectrum when the radius of these balls goes to infinity. This involves the spectrum, which we call macroscopic spectrum, of a so called homogenized operator on a specific domain. Furthermore we show that the first macroscopic eigenvalue is bounded from above, by a universal constant in the case of the three dimensional Heisenberg group, and by a constant depending on the Albanese torus for the other nilmanifolds. We also show that the Heisenberg groups belong to a family of nilmanifolds, where the equality characterizes some pseudo-left-invariant metrics.
A product formula for valuations on manifolds with applications to the integral geometry of the quaternionic line
Authors: Bernig Andreas
Published: December 2009

The Alesker–Poincaré pairing for smooth valuations on manifolds is expressed in terms of the Rumin differential operator acting on the cosphere-bundle. It is shown that the derivation operator, the signature operator and the Laplace operator acting on smooth valuations are formally self-adjoint with respect to this pairing. As an application, the product structure of the space of

\mathrm{SU}(2)

- and translation invariant valuations on the quaternionic line is described. The principal kinematic formula on the quaternionic line

ℍ

is stated and proved.
A Caporaso–Harris type formula for Welschinger invariants of real toric Del Pezzo surfaces
Authors: Itenberg Ilia , Kharlamov Viatcheslav , Shustin Eugenii
Published: December 2009

We define a series of relative tropical Welschinger-type invariants of real toric surfaces. In the Del Pezzo case, these invariants can be seen as real tropical analogs of relative Gromov–Witten invariants, and are subject to a recursive formula. As application we obtain new formulas for Welschinger invariants of real toric Del Pezzo surfaces.
String topology for spheres
Authors: Menichi Luc
Published: December 2009

Let

M

be a compact oriented

d

-dimensional smooth manifold. Chas and Sullivan have defined a structure of Batalin–Vilkovisky algebra on

ℍ_∗(LM)

. Extending work of Cohen, Jones and Yan, we compute this Batalin–Vilkovisky algebra structure when

M

is a sphere

S^d

,

d ≥1

. In particular, we show that

ℍ_∗(LS^2;\mathbb{F}_2)

and the Hochschild cohomology

HH^∗(H^∗(S^2);H^∗(S^2))

are surprisingly not isomorphic as Batalin–Vilkovisky algebras, although we prove that, as expected, the underlying Gerstenhaber algebras are isomorphic. The proof requires the knowledge of the Batalin–Vilkovisky algebra

H_∗(Ω^2S^3;\mathbb{F}_2)

that we compute in the Appendix.
$\underline{E}G$ for systolic groups
Authors: Przytycki Piotr
Published: December 2009

We prove that if a group

G

is systolic, i.e., if it acts properly and cocompactly on a systolic complex

X

, then an appropriate Rips complex constructed from

X

is a finite model for

\underline{E}G

.
Group splittings and integrality of traces
Authors: Emmanouil Ioannis
Published: December 2009

In this paper, we elaborate on Connes’ proof of the integrality of the trace conjecture for free groups, in order to show that any action of a group

G

on a tree leads to a similar integrality assertion concerning the trace on the group algebra

ℂG

, which is associated with the set of group elements that stabilize a vertex.
Freeness of conic-line arrangements in $ℙ^2$
Authors: Schenck Henry K. , Tohǎneanu Ştefan O.
Published: December 2009

Let

\mathcal C = \bigcup^n_{i = 1} C_i ⊆ ℙ^2

be a collection of smooth rational plane curves. We prove that the addition–deletion operation used in the study of hyperplane arrangements has an extension which works for a large class of arrangements of smooth rational curves, giving an inductive tool for understanding the freeness of the module

Ω^1(\mathcal C)

of logarithmic differential forms with pole along

\mathcal C

. We also show that the analog of Terao’s conjecture (freeness of

Ω^1(\mathcal C)

is combinatorially determined if

\mathcal C

is a union of lines) is false in this setting.
Amenable groups and Hadamard spaces with a totally disconnected isometry group
Authors: Caprace Pierre-Emmanuel
Published: December 2009

Let

X

be a locally compact Hadamard space and

G

be a totally disconnected group acting continuously, properly and cocompactly on

X

. We show that a closed subgroup of

G

is amenable if and only if it is (topologically locally finite)-by-(virtually abelian). We are led to consider a set

∂^{\mathrm{fine}}_∞  X

which is a refinement of the visual boundary

∂_∞ X

. For each

x \in ∂^{\mathrm{fine}}_∞  X

, the stabilizer

G_x

is amenable.
Quelques nouveaux phénomènes de rang 1 pour les groupes de difféomorphismes du cercle
Authors: Navas Andrés
Published: December 2009

Nous démontrons un théorème de super-rigidité pour les actions de réseaux de rang supérieur par difféomorphismes du cercle. We prove a superrigidity theorem for actions of higher rank lattices by diffeomorphisms of the circle.
Chern numbers and the geometry of partial flag manifolds
Authors: Kotschick Dieter , Terzić S.
Published: December 2009

We calculate the Chern classes and Chern numbers for the natural almost Hermitian structures of the partial flag manifolds

F_n = \mathit{SU}(n + 2)/S(U(n) × U(1) × U(1))

. For all

n > 1

there are two invariant complex algebraic structures, which arise from the projectivizations of the holomorphic tangent and cotangent bundles of

ℂP^{n + 1}

. The projectivization of the cotangent bundle is the twistor space of a Grassmannian considered as a quaternionic Kähler manifold. There is also an invariant nearly Kähler structure, because

F_n

is a 3-symmetric space. We explain the relations between the different structures and their Chern classes, and we prove that

F_n

is not geometrically formal.
Totally umbilic surfaces in homogeneous 3-manifolds
Authors: Souam Rabah , Toubiana Eric
Published: December 2009

We discuss existence and classification of totally umbilic surfaces in the model geometries of Thurston and the Berger spheres. We classify such surfaces in

ℍ^2 × ℝ

,

\mathbb{S}^2 × ℝ

and the

\mathit{Sol}

group. We prove nonexistence in the Berger spheres and in the remaining model geometries other than the space forms.
Some topological properties of quotients modulo semisimple algebraic groups
Authors: Gurjar R. V.
Published: December 2009

We will prove a general result in Invariant Theory, viz. for a quotient

ℂ^n/\!/G,

where

G

is a connected complex semisimple algebraic group, the local first homology group at any point in the quotient

ℂ^n/\!/G

is trivial and the local second homology group is finite. Using this we will prove that the completion of the local ring of any point in

ℂ^n/\!/G

is a unique factorization domain (UFD).
Topological rigidity and Gromov simplicial volume
Authors: Derbez Pierre
Published: December 2009

A natural problem in the theory of 3-manifolds is the question of whether two 3-manifolds are homeomorphic or not. The aim of this paper is to study this problem for the class of closed Haken manifolds using degree one maps.

To this purpose we introduce an invariant

τ(N) = (\mathrm{Vol}(N),\Vert N\Vert)

, where

\Vert N\Vert

denotes the Gromov simplicial volume of

N

and

\mathrm{Vol}(N)

is a 2-dimensional simplicial volume which measures the volume of the base 2-orbifolds of the Seifert pieces of

N

.

After studying the behavior of

τ(N)

under the action of non-zero degree maps, we prove that if

M

and

N

are closed Haken manifolds such that

\Vert M\Vert = |\mathrm{deg}(f)| \Vert N\Vert

and

\mathrm{Vol}(M) = \mathrm{Vol}(N)

then any non-zero degree map

f: M → N

is homotopic to a covering map. As a corollary we prove that if

M

and

N

are closed Haken manifolds such that

τ(N)

is sufficiently close to

τ(M)

then any degree one map

f: M → N

is homotopic to a homeomorphism.
Foliations and global inversion
Authors: Cabral Balreira Eduardo
Published: December 2009

We consider topological conditions under which a locally invertible map admits a global inverse. Our main theorem states that a local diffeomorphism

f: M → ℝ^n

is bijective if and only if

H_{n − 1}(M) = 0

and the pre-image of every affine hyperplane is non-empty and acyclic. The proof is based on some geometric constructions involving foliations and tools from intersection theory. This topological result generalizes in finite dimensions the classical analytic theorem of Hadamard–Plastock, including its recent improvement by Nollet–Xavier. The main theorem also relates to a conjecture of the aforementioned authors, involving the well-known Jacobian conjecture in algebraic geometry.
$\operatorname{SL}_n(ℤ[t])$ is not $\operatorname{FP}_{n− 1}$
Authors: Bux Kai-Uwe , Mohammadi Amir , Wortman Kevin
Published: December 2009

We prove the result from the title using the geometry of Euclidean buildings.
On the cohomology of generalized triangle groups
Authors: Ellis Graham , Williams Gerald
Published: December 2009

We describe a general approach to constructing small free

{\Bbb Z}\Gamma

-resolutions for certain infinite isometry groups

\Gamma

. We apply the method to a class of generalized triangle groups and use the resolution to compute the integral homology of these groups. In illustrating the method we also obtain resolutions for the classical triangle groups and for their infinite cyclic central extensions, considered previously by Strebel.
Points périodiques des fonctions rationnelles dans l'espace hyperbolique $p$-adique
Authors: Rivera-Letelier Juan
Published: December 2009

We study the dynamics of rational maps with coefficients in the field

{\Bbb C}_p

acting on the hyperbolic space

{\Bbb H}_p

. Our main result is that the number of periodic points in

{\Bbb H}_p

of such a rational map is either

0

,

1

or

\infty

, and we characterize those rational maps having precisely

0

or

1

periodic points. The main property we obtain is a criterion for the existence of infinitely many periodic points (of a special kind) in hyperbolic space. The proof of this criterion is analogous to G. Julia's proof of the density of repelling periodic points in the Julia set of a complex rational map.
A prime analogue of the Erdös--Pomerance conjecture for elliptic curves
Authors: Liu Yu-Ru
Published: December 2009

Let

E/{\mathbb Q}

be an elliptic curve of rank

\ge 1

and

b\in E({\mathbb Q})

a rational point of infinite order. For a prime

p

of good reduction, let

g_b(p)

be the order of the cyclic group generated by the reduction

\bar b

of

b

modulo

p

. We denote by

\omega(g_b(p))

the number of distinct prime divisors of

g_b(p)

. Assuming the GRH, we show that the normal order of

\omega(g_b(p))

is

\log \log p

. We also prove conditionally that there exists a normal distribution for the quantity

\frac{\omega(g_b(p)) - \log \log p}{\sqrt{\log \log p}}.

The latter result can be viewed as an elliptic analogue of a conjecture of Erdös and Pomerance about the distribution of

\omega(f_a(n))

, where

a

is a natural number

> 1

and

f_a(n)

the order of

a

modulo

n

.
The Weinstein conjecture for planar contact structures in dimension three
Authors: Abbas Casim , Cieliebak Kai , Hofer Helmut W.
Published: December 2009

In this paper we describe a general strategy for approaching the Weinstein conjecture in dimension three. We apply this approach to prove the Weinstein conjecture for a new class of contact manifolds (planar contact manifolds). We also discuss how the present approach reduces the general Weinstein conjecture in dimension three to a compactness problem for the solution set of a first order elliptic PDE.
Quasi-states and symplectic intersections
Authors: Entov Michael , Polterovich Leonid
Published: December 2009

We establish a link between symplectic topology and a recently emerg\-ed branch of functional analysis called the theory of quasi-states and quasi-measures (also known as topological measures). In the symplectic context quasi-states can be viewed as an algebraic way of packaging certain information contained in Floer theory, and in particular in spectral invariants of Hamiltonian diffeomorphisms introduced recently by Yong-Geun Oh. As a consequence we prove a number of new results on rigidity of intersections in symplectic manifolds. This work is a part of a joint project with Paul Biran.
Algebraic Reduction Theorem for complex codimension one singular foliations
Authors: Cerveau Dominique , Lins-Neto Alcides , Loray Frank , Pereira Jorge Vitório , Touzet Frédéric
Published: December 2009

Let

M

be a compact complex manifold equipped with

n=\dim(M)

meromorphic vector fields that are linearly independent at a generic point. The main theorem is the following. If

M

is not bimeromorphic to an algebraic manifold, then any codimension one complex foliation

\mathcal F

with a codimension

\ge2

singular set is the meromorphic pull-back of an algebraic foliation on a lower dimensional algebraic manifold, or

\mathcal F

is transversely projective outside a proper analytic subset. The two ingredients of the proof are the Algebraic Reduction Theorem for the complex manifold

M

and an algebraic version of Lie's first theorem which is due to J. Tits.
An inverse spectral problem on surfaces
Authors: Castillon Philippe
Published: December 2009

The purpose of this paper is to prove how the positivity of some operators on a Riemannian surface gives informations on the conformal type of the surface (the operators considered here are of the form

\Delta+\lambda\mathcal{K}

where

\Delta

is the Laplacian of the surface,

\mathcal{K}

is its curvature and

\lambda

is a real number). In particular we obtain a theorem “à la Huber”: under a spectral hypothesis we prove that the surface is conformally equivalent to a Riemann surface with a finite number of points removed. This problem has its origin in the study of stable minimal surfaces.
H-minimal graphs of low regularity in $\mathbb H^1$
Authors: Pauls Scott D.
Published: December 2009

In this paper we investigate H-minimal graphs of lower regularity. We show that noncharacteristic

C^1

H-minimal graphs whose components of the unit horizontal Gauss map are in

W^{1,1}

are ruled surfaces with

C^2

seed curves. Moreover, in light of a structure theorem of Franchi, Serapioni and Serra Cassano, we see that any H-minimal graph is, up to a set of perimeter zero, composed of such pieces. Along these lines, we investigate ways in which patches of

C^1

H-minimal graphs can be glued together to form continuous piecewise

C^1

H-minimal surfaces. We apply this description of H-minimal graphs to the question of the existence of smooth solutions to the Dirichlet problem with smooth data. We find a necessary and sufficient condition for the existence of smooth solutions and produce examples where the conditions are satisfied and where they fail. In particular we illustrate the failure of the smoothness of the data to force smoothness of the solution to the Dirichlet problem by producing a class of curves whose H-minimal spanning graphs cannot be

C^2

.
Uniqueness of constant mean curvature surfaces properly immersed in a slab
Authors: Alías Luis J. , Dajczer Marcos
Published: December 2009

We study complete properly immersed surfaces contained in a slab of a warped product

\mathbb{R}\times_\varrho\mathbb{P}^2

, where

\mathbb{P}^2

is complete with nonnegative Gaussian curvature. Under certain restrictions on the mean curvature of the surface we show that such an immersion does not exists or must be a leaf of the trivial totally umbilical foliation

t \in \mathbb{R}\mapsto \{t\} \times \mathbb{P}^2

.
Complete proper minimal surfaces in convex bodies of $\mathbb R^3$, II. The behavior of the limit set
Authors: Martín Francisco , Morales Santiago
Published: December 2009

Let

D

be a regular, strictly convex bounded domain of

\mathbb{R}^3

, and consider a Jordan curve

\Gamma \subset \partial D

. Then, for each

\varepsilon>0

, we obtain the existence of a complete proper minimal immersion

\psi_\varepsilon \colon \mathbb{D} \rightarrow D

satisfying that the Hausdorff distance

\delta^H(\psi_\varepsilon(\partial \mathbb{D}), \Gamma) < \varepsilon

, where

\psi_\varepsilon(\partial \mathbb{D})

represents the limit set of the minimal disk

\psi_\varepsilon(\mathbb{D})

.
An algorithm to find vertical tori in small Seifert fiber spaces
Authors: Li Tao
Published: December 2009

We give an algorithm to find vertical essential tori in small Seifert fiber spaces with infinite fundamental group. This implies that there are algorithms to decide whether a 3-manifold is a Seifert fiber space.
Existence of quasi-periodic solutions for elliptic equations on a cylindrical domain
Authors: Valls Claudia
Published: December 2009

The elliptic equation

\partial_{tt} u= -\partial_{xx} u - \alpha u - g(u)

,

\alpha >0

is ill-posed and "most'' initial conditions lead to no solutions. Nevertheless, we show that for almost every

\alpha

there exist smooth solutions which are quasi-periodic. These solutions are anti-symmetric in space, and hence they are not traveling waves. Our approach uses the existence of an invariant center manifold, and the solutions are obtained from a KAM-type theorem for the restriction of the equation to that manifold.
Sharp inequalities for the coefficients of concave schlicht functions
Authors: Avkhadiev Farit G. , Pommerenke Christian , Wirths Karl-Joachim
Published: December 2009

Let

D

denote the open unit disc and let

f\colon D\to \mathbb{C}

be holomorphic and injective in

D

. We further assume that

f(D)

is unbounded and

\mathbb{C}\setminus f(D)

is a convex domain. In this article, we consider the Taylor coefficients

a_n(f)

of the normalized expansion

f(z)=z+\sum_{n=2}^{\infty}a_n(f)z^n, z\in D,

and we impose on such functions

f

the second normalization

f(1)=\infty

. We call these functions concave schlicht functions, as the image of

D

is a concave domain. We prove that the sharp inequalities

|a_n(f)-\frac{n+1}{2}|\leq\frac{n-1}{2}, n\geq 2,

are valid. This settles a conjecture formulated in [2].
Metrics on diagram groups and uniform embeddings in a Hilbert space
Authors: Arzhantseva Goulnara N. , Guba V. S. , Sapir Mark V.
Published: December 2009

We give first examples of finitely generated groups having an intermediate, with values in

(0,1)

, Hilbert space compression (which is a numerical parameter measuring the distortion required to embed a metric space into Hilbert space). These groups include certain diagram groups. In particular, we show that the Hilbert space compression of Richard Thompson's group

F

is equal to

1/2

, the Hilbert space compression of

\mathbb{Z}\wr\mathbb{Z}

is between

1/2

and

3/4

, and the Hilbert space compression of

\mathbb{Z}\wr(\mathbb{Z}\wr\mathbb{Z})

is between 0 and

1/2

. In general, we find a relationship between the growth of

H

and the Hilbert space compression of

\mathbb{Z}\wr H

.
Permutation complexes for profinite groups
Authors: Symonds Peter
Published: December 2009

An important tool in the analysis of discrete groups of finite virtual cohomological dimension is the existence of a finite dimensional contractible CW-complex on which the group acts with finite stabilizers. We develop a purely algebraic analogue for profinite groups. This enables us to reveal the connection between finiteness conditions on the cohomology of the group and those on the normalizers of the finite p-subgroups.
Invariant d'Hermite isotrope et densité des réseaux orthogonaux lorentziens
Authors: Bavard Christophe
Published: December 2009

Nous déterminons la densité maximale des réseaux orthogonaux lorentziens jusqu'en dimension 12 et en dimension 18 pour le type pair. Par ailleurs, nous définissons un invariant d'Hermite isotrope pour lequel nous établissons, dans le cas lorentzien, une théorie «de Voronoï» complète.

We compute the maximal density of orthogonal Lorentzians lattices up to dimension 12 and in dimension 18 for the even type. On the other hand, we define an isotropic Hermite invariant and we show that it satisfies, in the Lorentzian case, a complete “Voronoï's theory” .
The barycenter method on singular spaces
Authors: Storm Peter
Published: December 2009

Compact convex cores with totally geodesic boundary are proven to uniquely minimize volume over all hyperbolic 3-manifolds in the same homotopy class. This solves a conjecture in Kleinian groups concerning acylindrical 3-manifolds. Closed hyperbolic manifolds are proven to uniquely minimize volume over all compact hyperbolic cone-manifolds in the same homotopy class with cone angles

≤2π

. Closed hyperbolic manifolds are proven to minimize volume over all compact Alexandrov spaces with curvature bounded below by

−1

in the same homotopy class. A version of the Besson–Courtois–Gallot theorem is proven for

n

-manifolds with boundary. The proofs extend the techniques of Besson–Courtois–Gallot.
Erratum to “The Huber theorem for non-compact conformally flat manifolds”
Authors: Carron Gilles , Herzlich Marc
Published: December 2009

An argument in our paper The Huber theorem for non-compact conformally flat manifolds [Comment. Math. Helv. 77 (2002), 192–220] was not justified. Using recent work by G. Tian and J. Viaclovsky, we show that our result holds true.
A short note on Dynkin groups and convergence groups
Authors: Yaman Asli
Published: December 2009

We prove that the family of discrete non-elementary Dynkin groups coincides with the family of non-elementary convergence groups.
Permutation groups, minimal degrees and quantum computing
Authors: Kempe Julia , Pyber László , Shalev Aner
Published: December 2009

We study permutation groups of given minimal degree without the classical primitivity assumption. We provide sharp upper bounds on the order of a permutation group

H ≤ S_n

of minimal degree

m

and on the number of its elements of any given support. These results contribute to the foundations of a non-commutative coding theory.

A main application of our results concerns the Hidden Subgroup Problem for

S_n

in quantum computing. We completely characterize the hidden subgroups of

S_n

that can be distinguished from identity with weak Quantum Fourier Sampling, showing that these are exactly the subgroups with bounded minimal degree. This implies that the weak standard method for

S_n

has no advantage whatsoever over classical exhaustive search.
Complements of the socle in monolithic groups
Authors: Lucchini Andrea , Menegazzo Federico , Morigi Marta
Published: December 2009

We show that if H is a finite group with a unique minimal normal subgroup N, which is not abelian, then the number of conjugacy classes of complements of N in H is strictly smaller than |N|.
Extremely primitive groups
Authors: Mann Avinoam , Praeger Cheryl E. , Seress Ákos
Published: December 2009

A primitive permutation group is called extremely primitive if a point stabilizer acts primitively on each of its orbits. We prove that finite extremely primitive groups are of affine type or almost simple. Moreover, we determine the affine type examples up to finitely many exceptions.
Automatically presented groups
Authors: Erschler Anna
Published: December 2009

We introduce the notions of automatically presented groups and piecewise automatically presented groups. We show that if G is a piecewise automatically presented group satisfying the property T of Kazhdan, then G is finite. We prove that if G is amenable and finitely presented, then G is virtually abelian. We give further restrictions for a group to be piecewise automatically presented and study properties of such groups. We also give examples of automatically presented groups.
Asymptotic dimension and uniform embeddings
Authors: Gal Światosław R.
Published: December 2009

We study uniform embeddings of metric spaces, which satisfy some asymptotic tameness conditions such as finite asymptotic dimension, finite Assouad–Nagata dimension, polynomial dimension growth or polynomial growth, into function spaces. 
We show how the
type function
of a space with finite asymptotic dimension estimates its Hilbert (or any

ℓ^p

-) compression. In particular, we show that the spaces of finite asymptotic dimension with
linear type
(spaces with finite Assouad–Nagata dimension) have compression rate equal to one. 
We show, without the extra assumption that the space has the doubling property (finite Assouad dimension), that a space with polynomial growth has polynomial dimension growth and compression rate equal to one.

The method used allows us to obtain a lower bound on the compression of the lamplighter group

ℤ ≀ ℤ

, which has infinite asymptotic dimension.
Morita classes in the homology of $\mathrm{Aut}(F_n)$ vanish after one stabilization
Authors: Conant James , Vogtmann Karen
Published: December 2009

There is a series of cycles in the rational homology of the groups

\mathrm{Out}(F_n)

, first discovered by S. Morita, which have an elementary description in terms of finite graphs. The first two of these give nontrivial homology classes, and it is conjectured that they are all nontrivial. These cycles have natural lifts to the homology of

\mathrm{Aut}(F_n)

, which is stably trivial by a recent result of Galatius. We show that in fact a single application of the stabilization map

\mathrm{Aut}(F_n) → \mathrm{Aut}(F_{n+1})

kills the Morita classes, so that they disappear immediately after they appear.
Minimal topological actions do not determine the measurable orbit equivalence class
Authors: Ceccherini-Silberstein Tullio , Elek Gábor
Published: December 2009

We construct an amenable action Φ of a non-amenable group Γ on a discrete space. This action extends to a minimal topological action Φ of Γ on a Cantor set C. We show that Φ is non-uniquely ergodic and furthermore there exist ergodic invariant measures μ1 and μ2 such that (Φ,C,μ1) and (Φ,C,μ2) are not orbit equivalent measurable equivalence relations. This also provides an instance of the failure of equivalence between the notions of “global” and “local” amenability for countable equivalence relations.
Symbolic dynamics and relatively hyperbolic groups
Authors: Dahmani François , Yaman Asli
Published: December 2009

We study the action of a relatively hyperbolic group on its boundary by methods of symbolic dynamics. We show that this dynamical system is expansive, and, under a condition on parabolic subgroups (satisfied in most examples), that it is finitely presented, meaning that it can be factorized through a subshift of finite type.
Natural central extensions of groups
Authors: Liedtke Christian
Published: December 2009

Given a group

G

and an integer

n ≥ 2

, we construct a new group

\tilde{\mathcal{K}}(G,n)

. Although this construction naturally occurs in the context of finding new invariants for complex algebraic surfaces, it is related to the theory of central extensions and the Schur multiplier. A surprising application is that Abelian groups of odd order possess naturally defined covers that can be computed from a given cover by a kind of warped Baer sum.
The weak hyperbolization conjecture for 3-dimensional CAT(0)-groups
Authors: Kapovich Michael , Kleiner Bruce
Published: December 2009

We prove a weak hyperbolization conjecture for CAT(0) 3-dimensional Poincaré duality groups.
The conjugacy problem for Coxeter groups
Authors: Krammer Daan
Published: December 2009

We prove that for every finite rank Coxeter group there exists a polynomial (cubic) solution to the conjugacy problem.
Examples of buildings constructed via covering spaces
Authors: Davis Michael W.
Published: December 2009

Covering space theory is used to construct new examples of buildings.
Low degree bounded cohomology and $L^2$-invariants for negatively curved groups
Authors: Thom Andreas
Published: December 2009

We study the subgroup structure of discrete groups that share cohomological properties which resemble non-negative curvature. Examples include all Gromov hyperbolic groups.

We provide strong restrictions on the possible s-normal subgroups of a ‘negatively curved’ group. Another result says that the image of a group, which is boundedly generated by a finite set of amenable subgroups, in a group, which admits a proper quasi-1-cocycle into the regular representation, has to be amenable. These results extend to a certain class of randomorphisms in the sense of Monod.
Automorphisms of two-dimensional RAAGS and partially symmetric automorphisms of free groups
Authors: Bux Kai-Uwe , Charney Ruth , Vogtmann Karen
Published: December 2009

We compute the virtual cohomological dimension of the group of partially symmetric outer automorphisms of a free group. We use this to obtain new upper and lower bounds on the VCD of the outer automorphism group of a

2

-dimensional right-angled Artin group. In the case of a right-angled Artin group with defining graph a tree, the bounds agree.
Anosov diffeomorphisms on infra-nilmanifolds modeled on a free 2-step nilpotent Lie group
Authors: Dekimpe Karel , Verheyen Kelly
Published: December 2009

In this article we study the existence question of Anosov diffeomorphisms on an infra-nilmanifold. After establishing a general existence criterion in terms of the associated holonomy representation, we concentrate on infra-nilmanifolds for which the covering Lie group is a free nilpotent Lie group. In turns out that in this case the criterion obtained before can be reduced drastically. Finally, we completely solve the existence question in case the covering Lie group is free 2-step nilpotent and the holonomy group is abelian.
Geometric characterization of flat groups of automorphisms
Authors: Baumgartner Udo , Schlichting Günter , Willis George A.
Published: December 2009

If

\mathcal H

is a flat group of automorphisms of finite rank

n

of a totally disconnected, locally compact group

G

, then each orbit of

\mathcal H

in the metric space

\mathcal B(G)

of compact, open subgroups of

G

is quasi-isometric to

n

-dimensional Euclidean space. In this note we prove the following partial converse: Assume that

G

is a totally disconnected, locally compact group such that

\mathcal B(G)

is a proper metric space and let

\mathcal H

be a group of automorphisms of

G

such that some (equivalently every) orbit of

\mathcal H

in

\mathcal B(G)

is quasi-isometric to

n

-dimensional Euclidean space, then

\mathcal H

has a finite index subgroup which is flat of rank

n

. We can draw this conclusion under weaker assumptions. We also single out a naturally defined flat subgroup of such groups of automorphisms.
Deformation spaces of trees
Authors: Guirardel Vincent , Levitt Gilbert
Published: December 2009

Let G be a finitely generated group. Two simplicial G-trees are said to be in the same deformation space if they have the same elliptic subgroups (if H fixes a point in one tree, it also does in the other). Examples include Culler–Vogtmann's outer space and spaces of JSJ decompositions. We discuss what features are common to trees in a given deformation space, how to pass from one tree to all other trees in its deformation space, and the topology of deformation spaces. In particular, we prove that all deformation spaces are contractible complexes.
Examples of hyperlinear groups without factorization property
Authors: Thom Andreas
Published: December 2009

In this note we give an example of a group which is locally embeddable into finite groups (in particular it is initially subamenable, sofic and hence hyperlinear) but does not have Kirchberg’s factorization property. This group provides also an example of a sofic Kazhdan group which is not residually finite, answering a question from [ES05]. We also give an example of a group which is not initially subamenable but hyperlinear. Finally, we point out a mistake in [Kir94], Corollary 1.2 (v)

⇒

(i), and [Kir93], Corollary 7.3 (iii), and provide an example of a group which does not have the factorization property and is still a subgroup of a connected finite-dimensional Lie group.
Stable discretization of scalar and constrained vectorial Perona–Malik equation
Authors: Bartels Sören , Prohl Andreas
Published: December 2009

We survey recent results on analysis and numerics of the scalar Perona–Malik
equation. A vectorial Perona–Malik equation is introduced
to evolve unit vector fields
for directional diffusion.
For both cases, scalar and vectorial,
fully discrete schemes are proposed which fulfill a discrete energy law,
and satisfy a discrete sphere constraint in the vectorial case. Computational
experiments are provided to illustrate quantitative behaviors, and compare
with scalar total variation flow and heat flow of

p

-harmonic maps.
On a free boundary problem modelling inductive-heating processes
Authors: Yin Hong-Ming
Published: December 2009

In this paper we study a free boundary problem describing a melting process by using induction heating. The mathematical model in one-space dimension consists of a coupled-parabolic system in each phase along with a non-equilibrium kinetic condition on the interface. By applying an energy estimate and a Campanato type of estimates, it is shown that the problem has a unique classical solution globally.
Variational models for peeling problems
Authors: Maddalena Francesco , Percivale Danilo
Published: December 2009

We study variational models for flexural beams and plates interacting with a rigid substrate through an adhesive layer. The general structure of the minimizers is investigated and some properties characterizing the system's dependence from the load and the material stiffnesses are discussed.
Viscosity solutions for a model of contact line motion
Authors: Glasner Karl , Kim Inwon C.
Published: December 2009

This paper considers a free boundary problem that describes the motion of contact lines of a liquid droplet on a flat surface. The elliptic nature of the equation for droplet shape and the monotonic dependence of contact line velocity on contact angle allows us to introduce a notion of “viscosity” solutions for this problem. Unlike similar free boundary problems, a comparison principle is only available for a modified short-time approximation because of the constraint that conserves volume. We use this modified problem to construct viscosity solutions to the original problem under a weak geometric restriction on the free boundary shape. We also prove uniqueness provided there is an upper bound on front velocity.
Regularity and nonexistence results for some free-interface problems related to Ginzburg–Landau vortices
Authors: Le Nam Q.
Published: December 2009

We study regularity and nonexistence properties for some free-interface problems arising in the study of limiting vorticities associated to the Ginzburg-Landau equations with magnetic field in two dimensions. Our results imply in particular that if these limiting vorticities concentrate on a smooth closed curve then they have a distinguished sign; moreover, if the domain is thin then solutions of the Ginzburg-Landau equations cannot have a number of vortices much larger than the applied magnetic field.
On the shape derivative for problems of Bernoulli type
Authors: Haslinger J. , Ito K. , Kozubek T. , Kunisch Karl , Peichl G.
Published: December 2009

The shape derivative of the cost functional in a Bernoulli-type problem is characterized. The calculation of the derivative of the cost does not use the shape derivative of the state variable and is achieved under mild regularity conditions on the boundary of the domain.
Linear stability of selfsimilar solutions of unstable thin-ﬁlm equations
Authors: Slepčev Dejan
Published: December 2009

We study the linear stability of selfsimilar solutions of long-wave unstable thin-ﬁlm equations with power-law nonlinearities

u_t = −(u^n u_{xxx} + u^m u_x)_x\qquad \text{for }0 < n < 3, \ n≤ m.

Steady states, which exist for all values of

m

and

n

above, are shown to be stable if

m ≤ n + 2

when

0 < n ≤ 2

, marginally stable if

m ≤ n + 2

when

2 < n < 3

, and unstable otherwise. Dynamical selfsimilar solutions are known to exist for a range of values of

n

when

m = n + 2

. We carry out the analysis of the stability of these solutions when

n= 1

and

m = 3

. Spreading selfsimilar solutions are proven to be stable. Selfsimilar blowup solutions with a single local maximum are proven to be stable, while selfsimilar blowup solutions with more than one local maximum are shown to be unstable.

The equations above are gradient ﬂows of a nonconvex energy on formal inﬁnite-dimensional manifolds. In the special case

n = 1

the equations are gradient ﬂows with respect to the Wasserstein metric. The geometric structure of the equations plays an important role in the analysis and provides a natural way to approach a family of linear stability problems.
On the energy of domain walls in ferromagnetism
Authors: Moser Roger
Published: December 2009

In the theory of micromagnetics, the magnetization of a ferromagnetic sample has an energy that is the sum of four components. We study the asymptotic behaviour of this functional when a parameter (the so-called exchange length) tends to 0. The interaction of two of the components of the energy permits the use of well-known methods from the theory of phase transitions. In the limit this gives rise to a division of the sample into domains of constant magnetization, separated by domain walls. We examine the contribution of a third energy (the energy of the stray ﬁeld) to the limiting problem. In particular, we derive an estimate for the energy density on the domain walls.
Uniform energy and density distribution: diblock copolymers’ functional
Authors: Nunzio Spadaro Emanuele
Published: December 2009

We study a nonlocal variational problem arising in diblock copolymers models, whose energy is given by the Cahn–Hilliard functional plus a long-range interaction term. We prove that minimizers develop uniform energy and density distributions, thus justifying partially the highly regular microphase separation observed in diblock copolymers’ melts. We also give a new proof of the scaling law for the minimum energy. This work extends the techniques introduced in [1] where analogous results are proved for the sharp interface limit of the functional considered.
Bounded cohomology and isometry groups of hyperbolic spaces
Authors: Hamenstädt Ursula
Published: December 2009

Let

X

be an arbitrary hyperbolic geodesic metric space and let

\Gamma

be a countable subgroup of the isometry group

{\rm Iso}(X)

of

X

. We show that if

\Gamma

is non-elementary and weakly acylindrical (this is a weak properness condition) then the second bounded cohomology groups

H_b^2(\Gamma,\mathbb{R})

,

H_b^2(\Gamma,\ell^p(\Gamma))

(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 product.