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European Mathematical Society Publishing House
2024-03-28 21:23:47
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https://www.ems-ph.org/meta/jmeta-stream.php?jrn=CMH&vol=83&iss=4&update_since=2024-03-28
Commentarii Mathematici Helvetici
Comment. Math. Helv.
CMH
0010-2571
1420-8946
General
10.4171/CMH
http://www.ems-ph.org/doi/10.4171/CMH
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European Mathematical Society Publishing House
Zuerich, Switzerland
© Swiss Mathematical Society
83
2008
4
On the appearance of Eisenstein series through degeneration
Daniel
Garbin
The Graduate Center of CUNY, NEW YORK, UNITED STATES
Jay
Jorgenson
The City College of New York - CUNY, NEW YORK, UNITED STATES
Michael
Munn
The Graduate Center of CUNY, NEW YORK, UNITED STATES
Hyperbolic Eisenstein series, degenerating Riemann surfaces, and counting functions
Let Γ be a Fuchsian group of the first kind acting on the hyperbolic upper half plane ℍ, and let M = Γ \ ℍ be the associated finite volume hyperbolic Riemann surface. If γ is parabolic, there is an associated (parabolic) Eisenstein series, which, by now, is a classical part of mathematical literature. If γ is hyperbolic, then, following ideas due to Kudla–Millson, there is a corresponding hyperbolic Eisenstein series. In this article, we study the limiting behavior of parabolic and hyperbolic Eisenstein series on a degenerating family of finite volume hyperbolic Riemann surfaces. In particular, we prove the following result. If γ ∈ Γ corresponds to a degenerating hyperbolic element, then a multiple of the associated hyperbolic Eisenstein series converges to parabolic Eisenstein series on the limit surface.
Number theory
Functions of a complex variable
General
701
721
10.4171/CMH/140
http://www.ems-ph.org/doi/10.4171/CMH/140
The rational homotopy Lie algebra of function spaces
Urtzi
Buijs
Universidad de Málaga, MÁLAGA, SPAIN
Aniceto
Murillo
Universidad de Málaga, MÁLAGA, SPAIN
Function space, homotopy Lie algebra, Sullivan model, rational homotopy theory
In this paper we fully describe the rational homotopy Lie algebra of any component of a given (free or pointed) function space. Also, we characterize higher order Whitehead products on these spaces. From this, we deduce the existence of H-structures on a given component of a pointed mapping space ℱ*(X,Y;f ) between rational spaces, assuming the cone length of X is smaller than the order of any non trivial generalized Whitehead product in π*(Y).
Algebraic topology
General topology
General
723
739
10.4171/CMH/141
http://www.ems-ph.org/doi/10.4171/CMH/141
Polynomial-time word problems
Saul
Schleimer
University of Warwick, COVENTRY, UNITED KINGDOM
Word problem, automorphisms, free groups, straight-line programs
We find polynomial-time solutions to the word problem for free-by-cyclic groups, the word problem for automorphism groups of free groups, and the membership problem for the handlebody subgroup of the mapping class group. All of these results follow from observing that automorphisms of the free group strongly resemble straight-line programs, which are widely studied in the theory of compressed data structures. In an effort to be self-contained we give a detailed exposition of the necessary results from computer science.
Group theory and generalizations
Manifolds and cell complexes
General
741
765
10.4171/CMH/142
http://www.ems-ph.org/doi/10.4171/CMH/142
Genus-one helicoids from a variational point of view
Donald
Ornstein
Stanford University, STANFORD, UNITED STATES
Brian
White
Stanford University, STANFORD, UNITED STATES
Complete embedded minimal surface, helicoid, variational methods
In this paper, we use variational methods to prove existence of a complete, properly embedded, genus-one minimal surface that is asymptotic to a helicoid at infinity. We also prove some new properties of such helicoid-like surfaces.
Differential geometry
Calculus of variations and optimal control; optimization
General
767
813
10.4171/CMH/143
http://www.ems-ph.org/doi/10.4171/CMH/143
Rigidité topologique sous l'hypothèse « entropie majorée » et applications
Guillemette
Reviron
Université de Montpellier II, MONTPELLIER CEDEX 5, FRANCE
Espaces métriques, entropie volumique, rigidité topologique, distance de Gromov–Hausdorff, précompacité, spectre des longueurs, revêtements
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.
General
815
846
10.4171/CMH/144
http://www.ems-ph.org/doi/10.4171/CMH/144
Irreducibly represented groups
Bachir
Bekka
Université de Rennes I, RENNES CEDEX, FRANCE
Pierre
de la Harpe
Université de Genève, GENÈVE 4, SWITZERLAND
Group representations, irreducible representations, faithful representations, infinite groups, von Neumann algebras
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 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 ℋ.
Topological groups, Lie groups
Group theory and generalizations
General
847
868
10.4171/CMH/145
http://www.ems-ph.org/doi/10.4171/CMH/145
Reparamétrisation universelle de familles f-analytiques de cycles et théorème de f-aplatissement géométrique
Daniel
Barlet
Université Henri Poincaré, VANDOEUVRE CEDEX, FRANCE
Analytic and meromorphic equivalence relations, cycles, geometric flattening, universal reparametrization, meromorphic quotients
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.
Several complex variables and analytic spaces
General
869
888
10.4171/CMH/146
http://www.ems-ph.org/doi/10.4171/CMH/146
Volume growth, curvature decay, and critical metrics
Gang
Tian
Princeton University, PRINCETON, UNITED STATES
Jeff
Viaclovsky
University of Wisconsin, MADISON, UNITED STATES
Anti-self-dual metrics, ALE spaces, curvature decay, ε-regularity, orbifold compactness, volume growth
We make some improvements to our previous results in [TV05a] and [TV05b]. First, we prove a version of our volume growth theorem which does not require any assumption on the first Betti number. Second, we show that our local regularity theorem only requires a lower volume growth assumption, not a full Sobolev constant bound. As an application of these results, we can weaken the assumptions of several of our theorems in [TV05a] and [TV05b].
Differential geometry
Global analysis, analysis on manifolds
General
889
911
10.4171/CMH/147
http://www.ems-ph.org/doi/10.4171/CMH/147
On the Kneser–Tits problem for triality forms
Gopal
Prasad
University of Michigan, ANN ARBOR, UNITED STATES
Kneser–Tits problem, triality forms
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.
Group theory and generalizations
General
913
925
10.4171/CMH/148
http://www.ems-ph.org/doi/10.4171/CMH/148
On manifolds of small degree
Paltin
Ionescu
, Bucharest, ROMANIA
Embedded projective manifold, small degree, rational manifold, Fano manifold, adjunction mapping
Let X ⊂ ℙn be a complex connected projective, non-degenerate, linearly normal manifold of degree d ≤ n. The main result of this paper is a classification of such manifolds. As a by-product of the classification it follows that these manifolds are either rational or Fano. In particular, they are simply connected (hence regular) and of negative Kodaira dimension. Moreover, easy examples show that d ≤ n is the best possible bound for such properties to hold true. The proof of our theorem makes essential use of the adjunction mapping and, in particular, the main result of [15] plays a crucial role in the argument.
Algebraic geometry
General
927
940
10.4171/CMH/149
http://www.ems-ph.org/doi/10.4171/CMH/149