bluuebunny/crossref_metadata_2025 huggingface dataset:
Embeddings of Danielewski surfaces in affine space
Authors: Dubouloz Adrien
Published: December 2009

We construct explicit embeddings of Danielewski surfaces [4] in affine spaces. The equations defining these embeddings are obtained from the

2\times2

minors of a matrix attached to a weighted rooted tree

\gamma

. We characterize those surfaces

S_{\gamma}

with a trivial Makar-Limanov invariant in terms of their associated trees. We prove that every Danielewski surface

S

with a nontrivial Makar-Limanov invariant admits a closed embedding in an affine space

\mathbb{A}_{k}^{n}

in such a way that every

\mathbb{G}_{a,k}

-action on

S

extends to an action on

\mathbb{A}^{n}

defined by a triangular derivation. We show that a Danielewski surface

S

with a trivial Makar-Limanov invariant and non-isomorphic to a hypersurface with equation

xz-P(y)=0

in

\mathbb{A}_{k}^{3}

admits nonconjugated algebraically independent

\mathbb{G}_{a,k}

-actions.
Intersection homology and Alexander modules of hypersurface complements
Authors: Maxim Laurentiu
Published: December 2009

Let

V

be a degree

d

, reduced hypersurface in

\mathbb{CP}^{n+1}

,

n \geq 1

, and fix a generic hyperplane,

H

. Denote by

\mathcal{U}

the (affine) hypersurface complement,

\mathbb{CP}^{n+1}-V \cup H

, and let

\mathcal{U}^c

be the infinite cyclic covering of

\mathcal{U}

corresponding to the kernel of the total linking number homomorphism. Using intersection homology theory, we give a new construction of the Alexander modules

H_i(\mathcal{U}^c;\mathbb{Q})

of the hypersurface complement and show that, if

i \leq n

, these are torsion over the ring of rational Laurent polynomials. We also obtain obstructions on the associated global polynomials: their zeros are roots of unity of order

d

and are entirely determined by the local topological information encoded by the link pairs of singular strata of a stratification of the pair

(\mathbb{CP}^{n+1},V)

. As an application, we give obstructions on the eigenvalues of monodromy operators associated to
Fiberwise localization and the cube theorem
Authors: Chataur David , Scherer Jérôme
Published: December 2009

In this paper we explain when it is possible to construct fiberwise localizations in model categories. For pointed spaces, the general idea is to decompose the total space of a fibration as a diagram over the category of simplices of the base and replace it by the localized diagram. This of course is not possible in an arbitrary category. We have thus to adapt another construction which heavily depends on Mather's cube theorem. Working with model categories in which the cube theorem holds, we propose a few equivalent conditions under which fiberwise nullifications exist. We show that these techniques apply to yield a fiberwise plus-construction for differential graded algebras over cofibrant operads.
Topological groups of Kac–Moody type, right-angled twinnings and their lattices
Authors: Rémy Bertrand , Ronan Mark
Published: December 2009

We construct new groups satisfying the combinatorial axioms of twin root data, by amalgamating spherical parabolic subgroups. The corresponding buildings have right-angled Coxeter groups as Weyl groups. The possibility of mixing several ground fields, not available for Kac–Moody groups, leads to strong non-linearity properties for the groups and some of their subgroups. We also discuss a completion procedure for groups with twin root data which is the starting point for defining a large family of topologically simple groups.
Fixed point sets of parabolic isometries of CAT(0)-spaces
Authors: Fujiwara Koji , Nagano Koichi , Shioya Takashi
Published: December 2009

We study the fixed point set in the ideal boundary of a parabolic isometry of a proper CAT(0)-space. We show that the radius of the fixed point set is at most

\pi/2

, and study its centers. As a consequence, we prove that the set of fixed points is contractible with respect to the Tits topology.
Finiteness in derived categories of local rings
Authors: Dwyer William G. , Greenlees John P. C. , Iyengar Srikanth B.
Published: December 2009

New homotopy invariant finiteness conditions on modules over commutative rings are introduced, and their properties are studied systematically. A number of finiteness results for classical homological invariants like flat dimension, injective dimension, and Gorenstein dimension, are established. It is proved that these specialize to give results concerning modules over complete intersection local rings. A noteworthy feature is the use of techniques based on thick subcategories of derived categories.
Bubbling location for $F$-harmonic maps and inhomogeneous Landau–Lifshitz equations
Authors: Najib Salah , Han Pigong
Published: December 2009

Let

f

be a positive smooth function on a closed Riemann surface

(M,g)

. The

f

-energy of a map

u

from

M

to a Riemannian manifold

(N,h)

is defined as

E_f(u)=\int_Mf|\nabla u|^2 dV_g.

In this paper, we will study the blow-up properties of Palais–Smale sequences for

E_f

. We will show that, if a Palais–Smale sequence is not compact, then it must blow up at some critical points of

f

. As a consequence, if an inhomogeneous Landau–Lifshitz system, i.e. a solution of

u_t=u\times\tau_f(u)+\tau_f(u), u: M\rightarrow S^2,

blows up at time

\infty

, then the blow-up points must be the critical points of

f

.
Cohomogeneity one hypersurfaces of Euclidean Spaces
Authors: Mercuri Francesco , Podestà Fabio , Seixas José A. P. , Tojeiro Ruy
Published: December 2009

We study isometric immersions

f: M^{n}\to \mathbb{R}^{n+1}

into Euclidean space of dimension

n+1

of a complete Riemannian manifold of dimension

n

on which a compact connected group of intrinsic isometries acts with principal orbits of codimension one. We give a complete classification if either

n\geq 3

and

M^n

is compact or if

n\geq 5

and the connected components of the flat part of

M^n

are bounded. We also provide several sufficient conditions for

f

to be a hypersurface of revolution.
EZ-structures and topological applications
Authors: Farrell F. Thomas , Lafont Jean-François
Published: December 2009

In this paper, we introduce the notion of an EZ-structure on a group, an equivariant version of the Z-structures introduced by Bestvina [4]. Examples of groups having an EZ-structure include (1) torsion free

\delta

-hyperbolic groups, and (2) torsion free

\mathrm{CAT}(0)

-groups.

Our first theorem shows that any group having an EZ-structure has an action by homeomorphisms on some

(\mathbb D^n, \Delta)

, where

n

is sufficiently large, and

\Delta

is a closed subset of

\partial \mathbb D^n=S^{n-1}

. The action has the property that it is proper and cocompact on

\mathbb D^n-\Delta

, and that if

K\subset \mathbb D^n-\Delta

is compact, that

\mathrm{diam}(gK)

tends to zero as

g\rightarrow \infty

. We call this property

(*_\Delta)

.

Our second theorem uses techniques of Farrell–Hsiang [8] to show that the Novikov conjecture holds for any torsion-free discrete group satisfying condition

(*_\Delta)

(giving a new proof that torsion-free

\delta

-hyperbolic and

\mathrm{CAT}(0)

groups satisfy the Novikov conjecture).

Our third theorem gives another application of our main result. We show how, in the case of a torsion-free

\delta

-hyperbolic group

\Gamma

, we can obtain a lower bound for the homotopy groups

\pi_n(\mathcal P(B\Gamma))

, where

\mathcal P(\cdot )

is the stable topological pseudo-isotopy functor.
Topological model for a class of complex Hénon mappings
Authors: Bonnot Sylvain
Published: December 2009

In order to describe the dynamics of the complex Hénon map

H_{a,c}\colon \begin{pmatrix}x\\y\end{pmatrix} \mapsto \begin{pmatrix}P_c(x)-ay\\x\end{pmatrix}

, where

P_c\colon z \mapsto z^2+c

has an attractive fixed point, we build a global topological model

(g,Y)

. In this model

Y

is the complement in

\mathbb{R}^4

of a cone over a solenoid lying in the unit 3-sphere, and

g\colon Y\rightarrow Y

is a map given in spherical coordinates by

g(r,\theta)=(r^2,\sigma(\theta))

, where

\sigma

is a solenoidal map of degree two. Then we prove the existence of a constant

\varepsilon>0

such that any Hénon map

H_{a,c}

with

0<|a|<\varepsilon

is conjugate to our model

(g,Y)

.
When does the associated graded Lie algebra of an arrangement group decompose?
Authors: Papadima Stefan , Suciu Alexander I.
Published: December 2009

Let

\mathcal{A}

be a complex hyperplane arrangement, with fundamental group

G

and holonomy Lie algebra

\mathfrak{H}

. Suppose

\mathfrak{H}_3

is a free abelian group of minimum possible rank, given the values the Möbius function

\mu\colon \mathcal{L}_2\to \mathbb{Z}

takes on the rank

2

flats of

\mathcal{A}

. Then the associated graded Lie algebra of

G

decomposes (in degrees

\ge 2

) as a direct product of free Lie algebras. In particular, the ranks of the lower central series quotients of the group are given by

\phi_r(G)=\sum _{X\in \mathcal{L}_2} \phi_r(F_{\mu(X)})

, for

r\ge 2

. We illustrate this new Lower Central Series formula with several families of examples.
Non-existence of $n$-dimensional $T$-embedded discs in $\mathbb R^{2n}$
Authors: Stojanovic Gordana , Tabachnikov Serge
Published: December 2009

We prove non-existence of

C^2

-smooth embeddings of

n

-dimensional discs to

\mathbb{R}^{2n}

such that the tangent spaces at distinct points are pairwise disjoint.
Gauss-Manin connections for arrangements, IV. Nonresonant eigenvalues
Authors: Cohen Daniel C. , Orlik Peter
Published: December 2009

An arrangement is a finite set of hyperplanes in a finite dimensional complex affine space. A complex rank one local system on the arrangement complement is determined by a set of complex weights for the hyperplanes. We study the Gauss-Manin connection for the moduli space of arrangements of fixed combinatorial type in the cohomology of the complement with coefficients in the local system determined by the weights. For nonresonant weights, we solve the eigenvalue problem for the endomorphisms arising in the

1

-form associated to the Gauss-Manin connection.
Non-existence of homogeneous Einstein metrics
Authors: Böhm Christoph
Published: December 2009

We show that there exist infinitely many simply connected compact prime homogeneous spaces

G/H

with infinite second homotopy group which do not admit

G

-invariant Einstein metrics.
CAT(0) and CAT(-1) dimensions of torsion free hyperbolic groups
Authors: Brady Noel , Crisp John
Published: December 2009

We show that a particular free-by-cyclic group G has CAT(0) dimension equal to 2, but CAT(-1) dimension equal to 3. Starting from a fixed presentation 2-complex we define a family of non-positively curved piecewise Euclidean “model” spaces for G, and show that whenever the group acts properly discontinuously by isometries on any proper 2-dimensional CAT(0) space X there exists a G-equivariant map from the universal cover of one of the model spaces to X which is locally isometric off the 0-skeleton and injective on vertex links.

From this we deduce bounds on the relative translation lengths of various elements of G acting on any such space X by first studying the geometry of the model spaces. By taking HNN-extensions of G we then produce an infinite family of 2-dimensional hyperbolic groups which do not act properly discontinuously by isometries on any proper CAT(0) metric space of dimension 2. This family includes a free-by-cyclic group with free kernel of rank 6.
Rigidity theory for matroids
Authors: Develin Mike , Martin Jeremy L , Reiner Victor
Published: December 2009

Combinatorial rigidity theory seeks to describe the rigidity or flexibility of bar-joint frameworks in Rd in terms of the structure of the underlying graph G. The goal of this article is to broaden the foundations of combinatorial rigidity theory by replacing G with an arbitrary representable matroid M. The ideas of rigidity independence and parallel independence, as well as Laman's and Recski's combinatorial characterizations of 2-dimensional rigidity for graphs, can naturally be extended to this wider setting. As we explain, many of these fundamental concepts really depend only on the matroid associated with G (or its Tutte polynomial), and have little to do with the special nature of graphic matroids or the field R.

Our main result is a “nesting theorem” relating the various kinds of independence. Immediate corollaries include generalizations of Laman's Theorem, as well as the equality of 2-rigidity and 2-parallel independence. A key tool in our study is the space of photos of M, a natural algebraic variety whose irreducibility is closely related to the notions of rigidity independence and parallel independence.

The number of points on this variety, when working over a finite field, turns out to be an interesting Tutte polynomial evaluation.
An infinite family of non-concordant knots having the same Seifert form
Authors: Kim Taehee
Published: December 2009

By a recent result of Livingston, it is known that if a knot has a prime power branched cyclic cover that is not a homology sphere, then there is an infinite family of non-concordant knots having the same Seifert form as the knot. In this paper, we extend this result to the full extent. We show that if the knot has nontrivial Alexander polynomial, then there exists an infinite family of non-concordant knots having the same Seifert form as the knot. As a corollary, no nontrivial Alexander polynomial determines a unique knot concordance class. We use Cochran-Orr-Teichner's recent result on the knot concordance group and Cheeger-Gromov's von Neumann

\rho

-invariants with their universal bound for a 3-manifold.
Pincement spectral en courbure de Ricci positive
Authors: Bertrand Jérôme
Published: December 2009

We show that for

n

-dimensional manifolds with

\mathrm{Ric} ≥ (n − 1)g

and for

k

in

\{1,\ldots ,n + 1\}

, the

k

-th eigenvalue for the Laplacian is close to

n

if and only if the manifold contains a subset which is Gromov–Hausdorff close to the sphere

\mathbb S^{k −1}

 . For

k = n + 1

, this gives a new proof of results of Colding and Petersen which show that the

(n + 1)

-th eigenvalue is close to

n

if and only if the manifold is Gromov–Hausdorff close to the

n

-sphere.

Dans cet article, nous démontrons que sur les variétés riemanniennes de dimension

n

vérifiant

\mathrm{Ric} ≥ (n − 1)g

et pour

k

dans

\{1,\ldots ,n + 1\}

, la ke valeur propre du laplacien est proche de

n

si et seulement si la variété contient une partie Gromov–Hausdorff proche de la sphère

\mathbb S^{k −1}

. Pour

k = n + 1

, nous obtenons une nouvelle preuve des résultats de Petersen et Colding qui montrent que pour de telles variétés, la

(n + 1)^{\text{e}}

valeur propre est proche de

n

si et seulement si la variété est Gromov–Hausdorff proche de la sphère de dimension

n

.
Complete hyperbolic Stein manifolds with prescribed automorphism groups
Authors: Kan Su-Jen
Published: December 2009

It is well known that the automorphism group of a hyperbolic manifold is a Lie group. Conversely, it is interesting to see whether or not any Lie group can be prescribed as the automorphism group of a certain complex manifold.

When the Lie group

G

is compact and connected, this problem has been completely solved by Bedford–Dadok and independently by Saerens–Zame in 1987. They have constructed strictly pseudoconvex bounded domains

Ω

such that

\mathrm{Aut}(Ω) = G

. For Bedford–Dadok’s

Ω

,

0 ≤ \mathrm{dim}_ℂΩ − \mathrm{dim}_ℝG ≤ 1

; for generic Saerens–Zame’s

Ω

,

\mathrm{dim}_ℂ Ω ≫ \mathrm{dim}_ℝG

.

J. Winkelmann has answered affirmatively to noncompact connected Lie groups in recent years. He showed there exist Stein complete hyperbolic manifolds

Ω

such that

\mathrm{Aut}(Ω)  = G

. In his construction, it is typical that

\mathrm{dim}_ℂ Ω ≫ \mathrm{dim}_ℝG

.

In this article, we tackle this problem from a different aspect. We prove that for any connected Lie group

G

(compact or noncompact), there exist complete hyperbolic Stein manifolds

Ω

such that

\mathrm{Aut}(Ω) = G

with

\mathrm{dim}_ℂΩ = \mathrm{dim}_ℝG

. Working on a natural complexification of the real-analytic manifold

G

, our construction of

Ω

is geometrically concrete and elementary in nature.
On some one parameter families of genus 2 algebraic curves and half twists
Authors: Silhol Robert
Published: December 2009

In this paper we show that for certain families of surfaces in genus 2 there is an action of

\mathrm{PSL}_2(ℤ)

that can be expressed very naturally both in terms of Fenchel–Nielsen coordinates for the surfaces and in terms of equations of the associated algebraic curves. We also show that one of these families coincides with the

\mathrm{SL}_2(ℝ)

orbit of the translation surface tiled by three squares and that the above

\mathrm{PSL}_2(ℤ)

action is exactly induced by the natural action of

\mathrm{SL}_2(ℝ)

on this orbit.
Arakelov theory of even orthogonal Grassmannians
Authors: Tamvakis Harry
Published: December 2009

We study the Arakelov intersection ring of the arithmetic scheme

\mathrm{OG}

which parametrizes maximal isotropic subspaces in an even dimensional vector space, equipped with the standard hyperbolic quadratic form. We give a presentation of the ring

\mathrm{CH}(\mathrm{\overline{OG}})

(when

\mathrm{OG}(ℂ)

is given its natural invariant hermitian metric) and formulate an ‘arithmetic Schubert calculus’ which extends the classical one for the cohomology ring of

\mathrm{OG}

. Our analysis leads to a computation of the Faltings height of

\mathrm{OG}

with respect to its fundamental embedding in projective space, and a comparison of the resulting formula with previous ones, due to Kaiser and Köhler [KK] and the author [T3], [T4].
A characterization of hyperbolic spaces
Authors: Chatterji Indira , Niblo Graham A.
Published: December 2009

We show that a geodesic metric space, and in particular the Cayley graph of a finitely generated group, is hyperbolic in the sense of Gromov if and only if intersections of any two metric balls is itself “almost” a metric ball. In particular,

\mathbb R

-trees are characterized among the class of geodesic metric spaces by the property that the intersection of any two metric balls is always a metric ball. A variation on the definition of “almost” allows us to characterise

\mathrm{CAT}(\kappa)

geometry for

\kappa \leq 0

in the same way.
The $\ell^2$-cohomology of hyperplane complements
Authors: Davis Michael W. , Januszkiewicz Tadeusz , Leary Ian J.
Published: December 2009

We compute the

\ell^2

-Betti numbers of the complement of a finite collection of affine hyperplanes in

{\mathbb C}^n

. At most one of the

\ell^2

-Betti numbers is nonzero.
Aperiodic colorings and tilings of Coxeter groups
Authors: Dranishnikov Alexander , Schroeder Viktor
Published: December 2009

We construct a limit aperiodic coloring of hyperbolic groups. Also we construct limit strongly aperiodic strictly balanced tilings of the Davis complex for all Coxeter groups.
The size of the solvable residual in finite groups
Authors: Dolfi Silvio , Herzog Marcel , Kaplan Gil , Lev Arieh
Published: December 2009

Let

G

be a finite group. The
solvable residual
of

G

, denoted by

\mathrm{Res}(G)

, is the smallest normal subgroup of

G

such that the respective quotient is solvable. We prove that every finite non-trivial group

G

with a trivial Fitting subgroup satisfies the inequality

|\mathrm{Res}(G)| > |G|^β

, where

β = \log(60)/\log(120(24)^{1/3}) ≈ 0.700265861

. The constant

β

in this inequality can not be replaced by a larger constant.
A partial extension of Lazard's correspondence for finite $p$-groups
Authors: Glauberman George
Published: December 2009

M. Lazard established a correspondence between finite

p

-groups of nilpotence class less than p and finite nilpotent Lie rings of

p

-power order and nilpotence class less than

p

. This correspondence has had many applications, but cannot generally be extended to

p

-groups of class

p

or larger. However, in this paper, we obtain a partial extension of Lazard's result for a

p

-group that is a product of normal subgroups of class less than

p

.
Magnus intersections of one-relator free products with small cancellation conditions
Authors: Juhász Arye
Published: December 2009

Donald Collins initiated the study of intersections of Magnus subgroups in one-relator groups. In particular, he characterized those intersections of Magnus subgroups that are not Magnus subgroups. In the present work we show that Collins’ results extend to one-relator quotients of free products of groups with a small cancellation condition and give a complete list of those defining relators for which Magnus subgroups do not intersect in a Magnus subgroup. We use van Kampen diagrams and word combinatorics.
Growth conditions in infinitely generated groups
Authors: Mann Avinoam
Published: December 2009

We characterize groups in which each finitely generated subgroup has polynomial growth, under some uniformity conditions.
Variations on a theme of Burns and Medvedev
Authors: Segal Dan
Published: December 2009

Burns and Medvedev prove in [BM] that a relatively free pro-p group cannot be p-adic analytic unless it is virtually nilpotent. We present a shorter, more conceptual proof, and apply it to deduce analogous results for other categories of groups.
Conjugacy in Garside groups II: structure of the ultra summit set
Authors: Birman Joan S. , Gebhardt Volker , González-Meneses Juan
Published: December 2009

This paper is the second in a series in which the authors study the conjugacy decision problem (CDP) and the conjugacy search problem (CSP) in Garside groups.

The ultra summit set

\mathrm{USS}(X)

of an element

X

in a Garside group

G

is a finite set of elements in

G

, which is a complete invariant of the conjugacy class of

X

in

G

. A fundamental question, if one wishes to find bounds on the size of

\mathrm{USS}(X)

, is to understand its structure. In this paper we introduce two new operations on elements

Y \in \mathrm{USS}(X)

, called ‘partial cycling’ and ‘partial twisted decycling’, and prove that if

Y, Z \in \mathrm{USS}(X)

, then

Y

and

Z

are related by sequences of partial cyclings and partial twisted decyclings. These operations are a concrete way to understand the minimal simple elements whose existence follows from the convexity property of ultra summit sets.

Using partial cycling and partial twisted decycling, we investigate the structure of a directed graph

Γ_X

determined by

\mathrm{USS}(X)

, and show that

Γ_X

can be decomposed into ‘black’ and ‘grey’ subgraphs. There are applications relating to the authors’ program for finding a polynomial solution to the CDP/CSP in the case of braids, which is outlined in the first paper of this series. A different application is to give a new algorithm for solving the CDP/CSP in Garside groups which is faster than all other known algorithms, even though its theoretical complexity is the same as that of the established algorithm using ultra summit sets. There are also applications to the theory of reductive groups.
A commutator description of the solvable radical of a finite group
Authors: Gordeev Nikolai , Grunewald Fritz , Kunyavskii Boris , Plotkin Eugene
Published: December 2009

We are looking for the smallest integer

k > 1

providing the following characterization of the solvable radical

R(G)

of any finite group

G: R(G)

coincides with the collection of all

g \in G

such that for any

k

elements

a_1, a_2, \ldots, a_k \in G

, the subgroup generated by the elements

g, a_iga_i^{−1}

,

i = 1, \ldots , k

, is solvable. We consider a similar problem of finding the smallest integer

ℓ > 1

with the property that

R(G)

coincides with the collection of all

g \in G

such that for any

ℓ

elements

b_1, b_2,\ldots , b_ℓ \in G

, the subgroup generated by the commutators

[g, bi]

,

i = 1, \ldots , ℓ

, is solvable. Conjecturally,

k = ℓ = 3

. We prove that both

k

and

ℓ

are at most

7

. In particular, this means that a finite group

G

is solvable if and only if every

8

conjugate elements of

G

generate a solvable subgroup.
The action of Thompson's group on a $\mathrm{CAT}(0)$ boundary
Authors: Farley Daniel
Published: December 2009

For a given locally finite

\mathrm{CAT}(0)

) cubical complex

X

with base vertex

\ast

, we define the
profile
of a given geodesic ray

c

issuing from

\ast

to be the collection of all hyperplanes (in the sense of Sageev) crossed by

c

. We give necessary conditions for a collection of hyperplanes to form the profile of a geodesic ray, and conjecture that these conditions are also sufficient.

We show that profiles in diagram and picture complexes can be expressed naturally as infinite pictures (or diagrams), and use this fact to describe the fixed points at infinity of the actions by Thompson's groups

F, T

, and

V

on their respective

\mathrm{CAT}(0)

cubical complexes. In particular, the actions of

T

and

V

have no global fixed points. We obtain a partial description of the fixed set of

F

; it consists, at least, of an arc

c

of Tits length

π/2

, and any other fixed points of

F

must have one particular profile, which we describe. We conjecture that all of the fixed points of

F

lie on the arc

c

.

Our results are motivated by the problem of determining whether

F

is amenable.
Kleinian groups with ubiquitous surface subgroups
Authors: Masters Joseph D.
Published: December 2009

We show that every finitely generated free subgroup of a right-angled, co-compact Kleinian reflection group is contained in a surface subgroup.
Iterated monodromy groups of quadratic polynomials, I
Authors: Bartholdi Laurent , Nekrashevych Volodymyr V.
Published: December 2009

We describe the iterated monodromy groups associated with post-critically finite quadratic polynomials, and make explicit their connection to the ‘kneading sequence’ of the polynomial.

We then give recursive presentations by generators and relations for these groups, and study some of their properties, like torsion and ‘branchness’.
Limits of Baumslag–Solitar groups
Authors: Guyot Luc , Stalder Yves
Published: December 2009

We give a parametrization by m-adic integers of the limits of Baumslag–Solitar groups (marked with a canonical set of generators). It is shown to be continuous and injective on the invertible m-adic integers. We show that all such limits are extensions of a free group by a lamplighter group and all but possibly one are not finitely presented. Finally, we give presentations related to natural actions on trees.
On group-theoretic models of randomness and genericity
Authors: Kapovich Ilya , Schupp Paul
Published: December 2009

We compare Gromov's density model of random groups with the Arzhantseva–Ol’shanskii model of genericity.
Entropy and isoperimetry for linear and non-linear group actions
Authors: Gromov Misha
Published: December 2009

We prove inequalities of isoperimetric type for groups acting on linear spaces and discuss related geometric and combinatorial problems, where we use the Boltzmann entropy to keep track of the cardinalities (and/or measures) of sets and of the dimensions of linear spaces.
Uniform non-amenability, cost, and the first $\ell^2$-Betti number
Authors: Lyons Russell , Pichot Mikaël , Vassout Stéphane
Published: December 2009

It is shown that 2

β

1(Γ) ≤

h

(Γ) for any countable group Γ, where β1(Γ) is the first ℓ2-Betti number and h(Γ) the uniform isoperimetric constant. In particular, a countable group with non-vanishing first ℓ2-Betti number is uniformly non-amenable.  We then define isoperimetric constants in the framework of measured equivalence relations. For an ergodic measured equivalence relation

R

of type II_1, the uniform isoperimetric constant

h

(

R

) of

R

is invariant under orbit equivalence and satisfies

2

β

_1(

R

) ≤ 2C(

R

) − 2 ≤

h

(

R

),

where

β

_1(

R

) is the first ℓ^2-Betti number and C(

R

) the cost of

R

in the sense of Levitt (in particular

h

(

R

) is a non-trivial invariant). In contrast with the group case, uniformly non-amenable measured equivalence relations of type II_1 always contain non-amenable subtreeings.  An ergodic version

h_e

(Γ) of the uniform isoperimetric constant

h

(Γ) is defined as the infimum over all essentially free ergodic and measure preserving actions α of Γ of the uniform isoperimetric constant

h

(

R_α

) of the equivalence relation

R_α

associated to

α

. By establishing a connection with the cost of measure-preserving equivalence relations, we prove that

h_e

(Γ) = 0 for any lattice Γ in a semi-simple Lie group of real rank at least 2 (while

h_e

(Γ) does not vanish in general).
On the first $L^p$-cohomology of discrete groups
Authors: Martin Florian , Valette Alain
Published: December 2009

For finitely generated groups

\Gamma

, the isomorphism between the first

\ell^{ p}

-cohomology

H^1_{(p)}(\Gamma)

and the reduced 1-cohomology with coefficients in

\ell^{ p}(\Gamma)

is exploited to obtain vanishing results for

H^1_{(p)}(\Gamma)

. The following cases are treated: groups acting on trees, groups with infinite center, wreath products, and lattices in product groups.
Deformation spaces of $G$-trees and automorphisms of Baumslag–Solitar groups
Authors: Clay Matt T.
Published: December 2009

We construct an invariant deformation retract of a deformation space of

G

-trees. We show that this complex is finite dimensional in certain cases and provide an example that is not finite dimensional. Using this complex we compute the automorphism group of the classical non-solvable Baumslag–Solitar groups BS(

p,q

). The most interesting case is when

p

properly divides

q

. Collins and Levin computed a presentation for Aut(BS(

p,q

)) in this case using algebraic methods. Our computation uses Bass–Serre theory to derive these presentations. Additionally, we provide a geometric argument showing Out(BS(

p,q

)) is not finitely generated when

p

properly divides

q

.
Linearisation of finite Abelian subgroups of the Cremona group of the plane
Authors: Blanc Jérémy
Published: December 2009

Given a finite Abelian subgroup of the Cremona group of the plane, we provide a way to decide whether it is birationally conjugate to a group of automorphisms of a minimal surface.

In particular, we prove that a finite cyclic group of birational transformations of the plane is linearisable if and only if none of its non-trivial elements fix a curve of positive genus. For finite Abelian groups, there exists only one surprising exception, a group isomorphic to

ℤ/2ℤ × ℤ/4ℤ

, whose non-trivial elements do not fix a curve of positive genus but which is not conjugate to a group of automorphisms of a minimal rational surface.

We also give some descriptions of automorphisms (not necessarily of finite order) of del Pezzo surfaces and conic bundles.
Local similarities and the Haagerup property (with an appendix by Daniel S. Farley)
Authors: Hughes Bruce
Published: December 2009

A new class of groups, the locally finitely determined groups of local similarities on compact ultrametric spaces, is introduced and it is proved that these groups have the Haagerup property (that is, they are a-T-menable in the sense of Gromov). The class includes Thompson's groups, which have already been shown to have the Haagerup property by D. S. Farley, as well as many other groups acting on boundaries of trees. A sufficient condition, used in this article, for the Haagerup property is shown in the appendix by D. S. Farley to be equivalent to the well-known property of having a proper action on a space with walls.
Denominator bounds in Thompson-like groups and flows
Authors: Calegari Danny
Published: December 2009

Let T denote Thompson's group of piecewise 2-adic linear homeomorphisms of the circle. Ghys and Sergiescu showed that the rotation number of every element of T is rational, but their proof is very indirect. We give here a short, direct proof using train tracks, which generalizes to elements of PL+(S1) with rational break points and derivatives which are powers of some fixed integer, and also to certain flows on surfaces which we call Thompson-like. We also obtain an explicit upper bound on the smallest period of a fixed point in terms of data which can be read off from the combinatorics of the homeomorphism.
Profinite completions of orientable Poincaré duality groups of dimension four and Euler characteristic zero
Authors: Kochloukova Dessislava H.
Published: December 2009

Let

p

be a prime number,

\mathcal T

be a class of finite groups closed under extensions, subgroups and quotients and the cyclic group of order

p

is in

\mathcal T

.

We find some sufficient and necessary conditions for the pro-

\mathcal T

completion of an abstract orientable Poincaré duality group

G

of dimension 4 and Euler characteristic 0 to be a profinite orientable Poincaré duality group of dimension 4 at the prime

p

with Euler

p

-characteristic 0. In particular we show that the pro-

p

completion

\widehat{G}_p

of

G

is an orientable Poincaré duality pro-

p

group of dimension 4 and Euler characteristic 0 if and only if

G

is

p

-good.
Periodic quotients of hyperbolic and large groups
Authors: Minasyan Ashot , Olshanskii Alexander Yu. , Sonkin Dmitriy
Published: December 2009

Let G be either a non-elementary (word) hyperbolic group or a large group (both in the sense of Gromov). In this article we describe several approaches for constructing continuous families of periodic quotients of G with various properties. The first three methods work for any non-elementary hyperbolic group, producing three different continua of periodic quotients of G. They are based on the results and techniques, that were developed by Ivanov and Olshanskii in order to show that there exists an integer n such that G/Gn is an infinite group of exponent n. The fourth approach starts with a large group G and produces a continuum of pairwise non-isomorphic periodic residually finite quotients. Speaking of a particular application, we use each of these methods to give a positive answer to a question of Wiegold from the Kourovka Notebook.
Flag-no-square triangulations and Gromov boundaries in dimension 3
Authors: Przytycki Piotr , Świątkowski Jacek
Published: December 2009

We describe an infinite family of 3-dimensional topological spaces, which are homeomorphic to boundaries of certain word-hyperbolic groups. The groups are right-angled hyperbolic Coxeter groups, whose nerves are flag-no-square triangulations of 3-dimensional manifolds. We prove that any 3-dimensional polyhedral complex (in particular, any 3-manifold) can be triangulated in a flag-no-square way.
On growth and torsion of groups
Authors: Bartholdi Laurent , Pochon Floriane
Published: December 2009

We give a subexponential upper bound and a superpolynomial lower bound on the growth function of the Fabrykowski–Gupta group.

As a consequence, we answer negatively a question by Longobardi, Maj and Rhemtulla about characterizing groups containing no free subsemigroups on two generators.
Représentation par des transvections des groupes d'Artin–Tits
Authors: Godelle Eddy
Published: December 2009

In a recent article, C. Kassel and C. Reutenauer studied the connection between the 4 strand braid group and Sturmian morphisms in word combinatorics. The aim of the current work is to extend this approach into a general connection between braid groups (of any index) and episturmian morphisms, a natural generalization of sturmian morphisms. Our key tool consists in associating with every graph a certain finite family of automorphisms of a free group. In the case of a complete graph, we recover some well-known family of episturmian morphisms. Now, considering the path of length n, we deduce a seemingly new representation of the braid group Bn +1 in Aut(Fn). By considering some other graphs, we similarly obtain representations of various Artin–Tits groups, in particular some affine braid groups. Our representation is faithful for B3 and B4; for other cases, the question of faithfulness remains open.
Free subalgebras of Lie algebras close to nilpotent
Authors: Belov Alexey , Mikhailov Roman
Published: December 2009

We prove that for every automata algebra of exponential growth the associated Lie algebra contains a free subalgebra. For

n ≥ 1

, let

L_{n + 2}

be a Lie algebra with generators

x_1,\ldots, x_{n + 2}

and the following relations: for

k ≤ n

, any commutator (with any arrangement of brackets) of length

k

which consists of fewer than

k

different symbols from

\{x_1, \ldots, x_{n + 2}\}

is zero. As an application of this result about automata algebras, we prove that

L_{n + 2}

contains a free subalgebra for every

n ≥ 1

. We also prove the similar result about groups defined by commutator relations. Let

G_{n + 2}

be a group with

n + 2

generators

y_1, \ldots, y_{n + 2}

and the following relations: for

k ≤ n

, any left-normalized commutator of length

k

which consists of fewer than

k

different symbols from

\{y_1, \ldots, y_{n + 2}\}

is trivial. Then the group

G_{n + 2}

contains a 2-generated free subgroup.

The main technical tool is combinatorics of words, namely combinatorics of periodical sequences and period switching.
Equivariant bundles and isotropy representations
Authors: Hambleton Ian , Hausmann Jean-Claude
Published: December 2009

We introduce a new construction, the
isotropy groupoid
, to organize the orbit data for split Γ-spaces. We show that equivariant principal G-bundles over split Γ-CW complexes X can be effectively classified by means of representations of their isotropy groupoids. For instance, if the quotient complex A = Γ \ X is a graph, with all edge stabilizers toral subgroups of Γ, we obtain a purely combinatorial classification of bundles with structural group G a compact connected Lie group. If G is abelian, our approach gives combinatorial and geometric descriptions of some results of Lashof–May–Segal [18] and Goresky–Kottwitz–MacPherson [10].
Describing all bi-orderings on Thompson’s group $F$
Authors: Navas Andrés , Rivas Cristóbal
Published: December 2009

We describe all possible ways of bi-ordering Thompson’s group

F

: its space of bi-orderings is made up of eight isolated points and four canonical copies of the Cantor set.
Traveling wave solutions of a nonlinear degenerate parabolic system from petroleum engineering
Authors: Bertsch Michiel , Nitsch Carlo
Published: December 2009

We study existence and qualitative properties of traveling wave solutions of a new free boundary problem which describes fluid flow in diatomite rocks. Diatomites are rather fragile and are characterized by a low permeability, which can increase due to the nonlocal accumulation of damage caused by the fluid flow. The traveling wave solutions give insight in the behavior near the free boundaries and show a strong parameter dependence. In particular we find in certain parameter ranges solutions with discontinuities across the free boundaries.
Convergent and conservative schemes for nonclassical solutions based on kinetic relations. I
Authors: Boutin Benjamin , Chalons Christophe , Lagoutière Frédéric , LeFloch Philippe G.
Published: December 2009

We propose a new numerical approach to compute nonclassical solutions to hyperbolic conservation laws. The class of finite difference schemes presented here is fully conservative and keep nonclassical shock waves as sharp interfaces, contrary to standard finite difference schemes. The main challenge is to achieve, at the discretization level, a consistency property with respect to a prescribed kinetic relation. The latter is required for the selection of physically meaningful nonclassical shocks. Our method is based on a reconstruction technique performed in each computational cell that may contain a nonclassical shock. To validate this approach, we establish several consistency and stability properties, and we perform careful numerical experiments. The convergence of the algorithm toward the physically meaningful solutions selected by a kinetic relation is demonstrated numerically for several test cases, including concave-convex as well as convex-concave flux-functions.
Properties of Sobolev-type metrics in the space of curves
Authors: Mennucci A.C.G. , Yezzi A. , Sundaramoorthi G.
Published: December 2009

We define a manifold

M

where objects

c\in M

are curves, which we parameterize as

c:S^1\to \mathbb R^n

(

n\ge 2

,

S^1

is the circle). We study geometries on the manifold of curves, provided by Sobolev-type Riemannian metrics

H^j

. These metrics have been shown to regularize gradient flows used in computer vision applications, see [13], [14], [16] and references therein.

We provide some basic results of

H^j

metrics; and, for the cases

j=1,2

, we characterize the completion of the space of smooth curves. We call these completions

H^1

and

H^2

Sobolev-type Riemannian Manifolds of Curves.” This result is fundamental since it is a first step in proving the existence of geodesics with respect to these metrics. As a byproduct, we prove that the Fréchet distance of curves (see [7]) coincides with the distance induced by the “Finsler

L^\infty

metric” defined in §2.2 of [18]
Long-time asymptotics of Hele–Shaw flow for perturbed balls with injection and suction
Authors: Vondenhoff E.
Published: December 2009

We discuss long-time behaviour of Hele-Shaw flow with injection and suction, for domains that are small perturbations of balls. An evolution equation for the motion of these domains is derived and linearised. We use spectral properties of the linearisation to show that in the case of injection, perturbations of balls decay algebraically. For classical Hele-Shaw flow, convergence turns out to be faster if low Richardson moments vanish. If for the three-dimensional case surface tension is included, all liquid can be removed by suction if the suction point and the geometric centre coincide and the ratio of suction speed and surface tension is small enough. An arbitrarily large portion of the liquid can be removed if the initial domain is sufficiently close to a ball. The main tools are the principle of linearised stability and the theory of H. Amann for abstract quasilinear parabolic evolution equations.
Traveling waves for the Keller–Segel system with Fisher birth terms
Authors: Nadin Gregoire , Perthame Benoît , Ryzhik Lenya
Published: December 2009

We consider the traveling wave problem for the one dimensional Keller-Segel system with a birth term of either a Fisher/KPP type or with a truncation for small population densities. We prove that there exists a solution under some stability conditions on the coefficients which enforce an upper bound on the solution and

\dot H^1(\R)

estimates. Solutions in the KPP case are built as a limit of traveling waves for the truncated birth rates (similar to ignition temperature in combustion theory).

We also discuss some general bounds and long time convergence for the solution of the Cauchy problem and in particular linear and nonlinear stability of the non-zero steady state.
Anisotropic mean curvature flow for two-dimensional surfaces in higher codimension: a numerical scheme
Authors: Pozzi Paola
Published: December 2009

We consider evolution of two dimensional parametric surfaces by ani\-sotropic mean curvature flow in

\R^n

, for arbitrary

n \geq 3

. After deriving a classical and a weak formulation of the flow, a fully discrete stable finite element scheme is proposed, and numerical tests and simulations are presented.
Phase boundary dynamics: transitions between ordered and disordered lipid monolayers
Authors: Alt Hans Wilhelm , Alt Wolfgang
Published: December 2009

Based on a general thermodynamical theory of mass and momentum, we propose and investigate a new phase field model for small transition layers between two spatially separated phases with intersecting free energy functions. We use a phase fraction, that only depends on the ratio of the two density components. From the phase field model we derive conditions for the sharp interface velocity and density jumps. The general model is motivated by and applied to the dynamics of lipid monolayers, which appear as surfactant on the strongly expanded and compressed thin water film of lung alveoli. While the \emph{liquid condensed ordered phase} (LC) of a flat lipid monolayer is characterized by high viscosity and limited compressibility, the \emph{liquid expanded disordered phase} (LE) is dominated by diffusion and high compressibility. In order to perform the asymptotic transition layer analysis at moving phase boundaries, a new nonlinear free energy interpolation model is proposed whose excess energy, in comparison to standard linear interpolations, contains an energy hump that has to be surpassed in a permissive transition from one phase to the other. This leads to a unique density jump condition in the case that the ordered phase is extending, whereas in the retracting case the jump densities are not restricted. The transition profiles and the resulting interface speed are numerically determined for a typical example by solving the nonlinear degenerate ODE-system. In a simplified 1-dimensional situation with low Reynolds number, the approximative macroscopic system of differential equations with moving sharp interface is numerically solved and interpreted in application to surfactant monolayers in lung alveoli.
A moving boundary problem for periodic Stokesian Hele–Shaw flows
Authors: Escher Joachim , Matioc Bogdan-Vasile
Published: December 2009

This paper is concerned with the motion of an incompressible, viscous fluid in a Hele-Shaw cell. The free surface is moving under the influence of gravity and the fluid is modelled using a modified Darcy law for Stokesian fluids. We combine results from the theory of quasilinear elliptic equations, analytic semigroups and Fourier multipliers to prove existence of a unique classical solution to the corresponding moving boundary problem.
Flat free boundaries regularity in two-phase problems for a class of fully nonlinear elliptic operators with variable coefﬁcients
Authors: Argiolas Roberto , Ferrari Fausto
Published: December 2009

We study the regularity of the free boundary in two-phase problems for fully nonlinear elliptic operators. In particular, we prove that ﬂat free boundaries are _C_1,γ.
Partial $L^1$ Monge–Kantorovich problem: variational formulation and numerical approximation
Authors: Barrett John W. , Prigozhin Leonid
Published: December 2009

We consider the Monge–Kantorovich problem with transportation cost equal to distance and a relaxed mass balance condition: instead of optimally transporting one given distribution of mass onto another with the same total mass, only a given amount of mass, m, has to be optimally transported. In this partial problem the given distributions are allowed to have different total masses and m should not exceed the least of them. We derive and analyze a variational formulation of the arising free boundary problem in optimal transportation. Furthermore, we introduce and analyse the ﬁnite element approximation of this formulation using the lowest order Raviart–Thomas element. Finally, we present some numerical experiments where both approximations to the optimal transportation domains and the optimal transport between them are computed.
Two-phase ﬂows involving capillary barriers in heterogeneous porous media
Authors: Cancès Clément , Gallouët Thierry , Porretta Alessio
Published: December 2009

We consider a simpliﬁed model of a two-phase ﬂow through a heterogeneous porous medium, in which convection is neglected. This leads to a nonlinear degenerate parabolic problem in a domain divided into an arbitrary ﬁnite number of homogeneous porous media. We introduce a new way to connect capillary pressures on the interfaces between the homogeneous domains, which leads to a general notion of solution. We then compare this notion of solution with an existing one, showing that it allows one to deal with a larger class of problems. We prove the existence of such a solution in a general case, and the existence and uniqueness of a regular solution in the one-dimensional case for initial data regular enough.