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European Mathematical Society Publishing House
2024-03-28 12:06:39
8
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=CMH&vol=82&iss=1&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
subscribers, moving wall 5 years
European Mathematical Society Publishing House
Zuerich, Switzerland
© Swiss Mathematical Society
82
2007
1
Permutation complexes for profinite groups
Peter
Symonds
University of Manchester, MANCHESTER, UNITED KINGDOM
Profinite group, cohomology, permutation module
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.
Group theory and generalizations
General
1
37
10.4171/CMH/83
http://www.ems-ph.org/doi/10.4171/CMH/83
Invariant d'Hermite isotrope et densité des réseaux orthogonaux lorentziens
Christophe
Bavard
Université Bordeaux, Talence Cedex, FRANCE
Réseaux euclidiens orthogonaux, invariant d'Hermite
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” .
Number theory
General
39
60
10.4171/CMH/84
http://www.ems-ph.org/doi/10.4171/CMH/84
CAT(0) and CAT(-1) dimensions of torsion free hyperbolic groups
Noel
Brady
University of Oklahoma, NORMAN, UNITED STATES
John
Crisp
Université de Bourgogne, DIJON CEDEX, FRANCE
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.
Group theory and generalizations
Differential geometry
Manifolds and cell complexes
General
61
85
10.4171/CMH/85
http://www.ems-ph.org/doi/10.4171/CMH/85
Isometric immersions into 3-dimensional homogeneous manifolds
Benoît
Daniel
, RIO DE JANEIRO, BRAZIL
Isometric immersions, constant mean curvature surfaces, homogeneous manifolds, Gauss and Codazzi equations
We give a necessary and sufficient condition for a 2-dimensional Riemannian manifold to be locally isometrically immersed into a 3-dimensional homogeneous Riemannian manifold with a 4-dimensional isometry group. The condition is expressed in terms of the metric, the second fundamental form, and data arising from an ambient Killing field. This class of 3-manifolds includes in particular the Berger spheres, the Heisenberg group Nil3, the universal cover of the Lie group PSL2(R) and the product spaces S2×R and H2×R. We give some applications to constant mean curvature (CMC) surfaces in these manifolds; in particular we prove the existence of a generalized Lawson correspondence, i.e., a local isometric correspondence between CMC surfaces in homogeneous 3-manifolds.
Differential geometry
General
87
131
10.4171/CMH/86
http://www.ems-ph.org/doi/10.4171/CMH/86
The barycenter method on singular spaces
Peter
Storm
University of Pennsylvania, PHILADELPHIA, UNITED STATES
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.
Differential geometry
Manifolds and cell complexes
Global analysis, analysis on manifolds
General
133
173
10.4171/CMH/87
http://www.ems-ph.org/doi/10.4171/CMH/87
A pinching theorem for the first eigenvalue of the Laplacian on hypersurfaces of the Euclidean space
Bruno
Colbois
Université de Neuchâtel, NEUCHÂTEL, SWITZERLAND
Jean-Francois
Grosjean
Université Henri Poincaré, VANDOEUVRE LES NANCY, FRANCE
Spectrum, Laplacian, pinching results, hypersurfaces
In this paper, we give pinching theorems for the first nonzero eigenvalue λ1(M) of the Laplacian on the compact hypersurfaces of the Euclidean space. Indeed, we prove that if the volume of M is 1 then, for any ε > 0, there exists a constant Cε depending on the dimension n of M and the L∞-norm of the mean curvature H, so that if the L2p-norm ||H||2p (p ≥ 2) of H satisfies n||H ||2p2 − Cε < λ1(M), then the Hausdorff-distance between M and a round sphere of radius (n/λ1(M))1/2 is smaller than ε. Furthermore, we prove that if C is a small enough constant depending on n and the L∞-norm of the second fundamental form, then the pinching condition n||H ||2p2 − C < λ1(M) implies that M is diffeomorphic to an n-dimensional sphere.
Differential geometry
General
175
195
10.4171/CMH/88
http://www.ems-ph.org/doi/10.4171/CMH/88
Rigidity theory for matroids
Mike
Develin
American Institute of Mathematics, PALO ALTO, UNITED STATES
Jeremy
Martin
University of Kansas, LAWRENCE, UNITED STATES
Victor
Reiner
University of Minnesota, MINNEAPOLIS, UNITED STATES
Matroid, combinatorial rigidity, parallel redrawing, Laman's Theorem, Tutte polynomial
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.
Combinatorics
Algebraic geometry
Convex and discrete geometry
General
197
233
10.4171/CMH/89
http://www.ems-ph.org/doi/10.4171/CMH/89
Erratum to "The topology at infinity of Coxeter groups and buildings"
Michael
Davis
Ohio State University, COLUMBUS, UNITED STATES
John
Meier
Lafayette College, EASTON, UNITED STATES
Group theory and generalizations
Manifolds and cell complexes
General
235
236
10.4171/CMH/90
http://www.ems-ph.org/doi/10.4171/CMH/90