- journal articles metadata
European Mathematical Society Publishing House
2024-03-29 14:27:35
8
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=LEM&vol=59&iss=1&update_since=2024-03-29
L’Enseignement Mathématique
Enseign. Math.
LEM
0013-8584
2309-4672
General
10.4171/LEM
http://www.ems-ph.org/doi/10.4171/LEM
subscribers
European Mathematical Society Publishing House
Zuerich, Switzerland
© Fondation L’Enseignement Mathématique
59
2013
1
Une preuve plus immobilière du théorème de «saturation» de Kapovich–Leeb–Millson
Nicole
Bardy-Panse
Université de Lorraine, CNRS, VANDOEUVRE LÈS NANCY, FRANCE
Cyril
Charignon
Université de Lorraine, CNRS, VANDOEUVRE LÈS NANCY, FRANCE
Stéphane
Gaussent
Université de Lorraine, CNRS, VANDOEUVRE LÈS NANCY, FRANCE
Guy
Rousseau
Université de Lorraine, CNRS, VANDOEUVRE LÈS NANCY, FRANCE
We give a more building-oriented and somewhat simpler proof of the « saturation » theorem of Kapovich and Millson for any complex semisimple group. The main difference with their approach lies in the combinatorial part of the proof. We state a theorem of folding/unfolding triangles in the affine building, only in combinatorial terms. For the analytical part, we gather materials that appear in distinct papers of Kapovich, Leeb and Millson to complete the proof.
General
3
37
10.4171/LEM/59-1-1
http://www.ems-ph.org/doi/10.4171/LEM/59-1-1
Le produit harmonique des suites
Bernard
Candelpergher
Université de Nice Sophia Antipolis, NICE CEDEX 2, FRANCE
Marc-Antoine
Coppo
Université de Nice Sophia Antipolis, NICE CEDEX 2, FRANCE
By means of an involutary binomial transformation on complex sequences, we define a new product called “harmonic” because of its remarkable properties towards harmonic sums. The Euler series transformation allows one to deduce from these properties some new and remarkable identities.
General
39
72
10.4171/LEM/59-1-2
http://www.ems-ph.org/doi/10.4171/LEM/59-1-2
Triangle groups, automorphic forms, and torus knots
Valdemar
Tsanov
Ruhr-Universität Bochum, BOCHUM, GERMANY
This article is concerned with the relation between several classical and well-known objects: triangle Fuchsian groups, $\mathbb{C}^\times$-equivariant singularities of plane curves, torus knot complements in the 3-sphere. The prototypical example is the modular group $PSL_2(\mathbb Z)$: the quotient of the nonzero tangent bundle on the upper-half plane by the action of $PSL_2(\mathbb Z)$ is biholomorphic to the complement of the plane curve $z^3-27w^2=0$. This can be shown using the fact that the algebra of modular forms is doubly generated, by $g_2,g_3$, and the cusp form $\Delta=g_2^3-27g_3^2$ does not vanish on the half-plane. As a byproduct, one finds a diffeomorphism between $PSL_2(\mathbb R)/PSL_2(\mathbb Z)$ and the complement of the trefoil knot - the local knot of the singular curve. This construction is generalized to include all $(p,q,\infty)$-triangle groups and, respectively, curves of the form $z^q+w^p=0$ and $(p,q)$-torus knots, for $p,q$ co-prime. The general case requires the use of automorphic forms on the simply connected group $\widetilde{SL_2}(\mathbb R)$. The proof uses ideas of Milnor and Dolgachev, which they introduced in their studies of the spectra of the algebras of automorphic forms of cocompact triangle groups (and, more generally, uniform lattices). It turns out that the same approach, with some modifications, allows one to handle the cuspidal case.
General
73
113
10.4171/LEM/59-1-3
http://www.ems-ph.org/doi/10.4171/LEM/59-1-3
How to turn a tetrahedron into a cube and similar transformations
G.C.
Shephard
East Anglia University, NORWICH, UNITED KINGDOM
Suppose that the surface of a polyhedron $P_{1}$ is cut in such a way that it can be opened out flat to form a connected region $R$ in the plane and that $R$, by introducing suitable folds, can be made into the net of a polyhedron $P_{2}$. Then we write $P_{1} \Rightarrow P_{2}$ and say that $P_{1}$ is transformed into $P_{2}$. In this paper we give many examples of the transformation of polyhedra and investigate the properties of the relation $ \Rightarrow $.
General
115
131
10.4171/LEM/59-1-4
http://www.ems-ph.org/doi/10.4171/LEM/59-1-4
Geometric covering arguments and ergodic theorems for free groups
Lewis
Bowen
The University of Texas at Austin, AUSTIN, UNITED STATES
Amos
Nevo
Technion - Israel Institute of Technology, HAIFA, ISRAEL
We present a new approach to the proof of ergodic theorems for actions of non-amenable groups, and give here a complete self-contained account of it in the case of free groups. Our approach is based on direct geometric covering arguments and asymptotic invariance arguments generalizing those developed in the ergodic theory of amenable groups. The results we describe go beyond those previously established for measure-preserving actions of free groups, and demonstrate the significant role the boundary action of the free group plays in the ergodic theory of its measure-preserving actions. Furthermore, our approach suggests the possibility of putting the ergodic theory of amenable groups and nonamenable groups on an equal footing : both can be viewed as special cases in the general ergodic theory of amenable ergodic equivalence relations with finite invariant measure.
General
133
164
10.4171/LEM/59-1-5
http://www.ems-ph.org/doi/10.4171/LEM/59-1-5
Projections and relative hyperbolicity
Alessandro
Sisto
Oxford University, OXFORD, UNITED KINGDOM
We give an alternative definition of relative hyperbolicity based on properties of closest-point projections on peripheral subgroups. We also derive a distance formula for relatively hyperbolic groups, similar to the one for mapping class groups.
General
165
181
10.4171/LEM/59-1-6
http://www.ems-ph.org/doi/10.4171/LEM/59-1-6
Almost-periodic action on the real line
Bertrand
Deroin
Université Paris-Sud, ORSAY CEDEX, FRANCE
A homeomorphism of the real line is almost-periodic if the set of its conjugates by the translations is relatively compact in the compact open topology. Our main result states that an action of a finitely generated group on the real line without global fixed points is conjugated to an action by almost-periodic homeomorphisms without almost fixed points. This is equivalent to saying that the real line together with the translation flow can be compactified as an orbit of a free action of $\mathbb R$ on a compact space, together with an action of the group by homeomorphisms withou global fixed points. As an application we give an alternative proof of Witte’s theorem: an amenable left orderable group is locally indicable.
General
183
194
10.4171/LEM/59-1-7
http://www.ems-ph.org/doi/10.4171/LEM/59-1-7
Limits of finite homogeneous metric spaces
Tsachik
Gelander
The Hebrew University of Jerusalem, JERUSALEM, ISRAEL
General
195
206
10.4171/LEM/59-1-8
http://www.ems-ph.org/doi/10.4171/LEM/59-1-8