Any Banach space has an equivalent norm with trivial isometries

For any Banach spaceX there is a norm |||·||| onX, equivalent to the original one, such that (X, |||·|||) has only trivial isometries. For any groupG there is a Banach spaceX such that the group of isometries ofX is isomorphic toG × {− 1, 1}. For any countable groupG there is a norm ‖ · ‖ _G onC([0, 1]) equivalent to the original one such that the group of isometries of (C([0, 1]), ‖ · ‖ _G ) is isomorphic toG × {−1, + 1}.     

The group of isometries of a locally compact metric space with one end

In this note we study the dynamics of the natural evaluation action of the group of isometries G of a locally compact metric space (X,d) with one end. Using the notion of pseudo-components introduced by S. Gao and A.S. Kechris we show that X has only finitely many pseudo-components exactly one of which is not compact and G acts properly on this pseudo-component. The complement of the non-compact component is a compact subset of X and G may fail to act properly on it.     

Isometry Groups of Proper Hyperbolic Spaces

Let X be a proper hyperbolic geodesic metric space and let G be a closed subgroup of the isometry group Iso(X) of X. We show that if G is not elementary then for every p ∈ (1, ∞) the second continuous bounded cohomology group H²_cb(G, L^p(G)) does not vanish. As an application, we derive some structure results for closed subgroups of Iso(X). Partially supported by Sonderforschungsbereich 611.     

Rank-One Isometries of Buildings and Quasi-Morphisms of Kac–Moody Groups

Given an irreducible non-spherical non-affine (possibly non-proper) building X, we give sufficient conditions for a group G < Aut(X) to admit an infinite-dimensional space of non-trivial quasi-morphisms. The result applies in particular to all irreducible (non-spherical and non-affine) Kac–Moody groups over integral domains. In particular, we obtain finitely presented simple groups of infinite commutator width, thereby answering a question of Valerii G. Bardakov [MK, Prob. 14.13]. Independently of these considerations, we also include a discussion of rank-one isometries of proper CAT(0) spaces from a rigidity viewpoint. In an appendix, we show that any homogeneous quasi-morphism of a locally compact group with integer values is continuous.     

Group Representations with Empty Residual Spectrum

Let X be a Banach space on which a discrete group Γ acts by isometries. For certain natural choices of X, every element of the group algebra, when regarded as an operator on X, has empty residual spectrum. We show, for instance, that this occurs if X is ℓ ²(Γ) or the group von Neumann algebra VN(Γ). In our approach, we introduce the notion of a surjunctive pair, and develop some of the basic properties of this construction. The cases X = ℓ ^p (Γ) for 1 ≤ p < 2 or 2 < p < ∞ are more difficult. If Γ is amenable we can obtain partial results, using a majorization result of Herz; an example of Willis shows that some condition on Γ is necessary.     

Multiple Solutions for a Class of Hemivariational Inequalities Involving Periodic Energy Functionals

In this paper we prove firstly that if f:X→ℝ is a locally Lipschitz function, bounded from below and invariant to a discrete group of dimension N is a suitable sense, acting on a Banach space X, then the problem: find u∈X such that o∈∂ f(u) (here ∂f(u) denotes Clarke's generalized gradient of f at x) admits at least N+1 orbits of solutions. Then, for a class of discrete groups G of isometries of a Hilbert space X, we establish an existence result for infinitely many G-orbits of eigensolutions to the problem: find u∈X such that λΛu∈∂f(u) for some λ∈ℝ, where Λ:X→X* stands for the duality map. The last two sections are devoted to applications of the abstract existence results to hemivariational inequalities possessing invariance properties. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Soluble minimal non-(finite-by-Baer)-groups

Let \mathfrakX{\mathfrak{X}} be a class of groups. A group G is called a minimal non- \mathfrakX{\mathfrak{X}}-group if it is not an \mathfrakX{\mathfrak{X}}-group but all of whose proper subgroups are \mathfrakX{\mathfrak{X}}-groups. In [16], Xu proved that if G is a soluble minimal non-Baer-group, then G/G ^′′ is a minimal non-nilpotent-group which possesses a maximal subgroup. In the present note, we prove that if G is a soluble minimal non-(finite-by-Baer)-group, then for all integer n ≥ 2, G/γ _n (G′) is a minimal non-(finite-by-abelian)-group.     

On Finite Groups and the Small Square Property

Let G be a finite group written multiplicatively and k a positive integer. If X is a non-empty subset of G, write X ² = |xy | x, y X . We say that G has the small square property on k-sets if |X ²| < k ² for any k-element subset X of G. For each group G, there is a unique m = m _G such that G has the small square property on (m + 1)-sets but not on m-sets. In this paper we show that given any positive integer d, there is a finite group G with m _G = d.     

Bounds on smooth matrix coefficients on L 2-spaces

Let G be a semisimple Lie group with a finite number of connected components and a finite center. Let K be a maximal compact subgroup. Let X be a smooth G-space equipped with a G-invariant measure. In this paper, we give upper bounds for K-finite and ${\mathfrak k}Let G be a semisimple Lie group with a finite number of connected components and a finite center. Let K be a maximal compact subgroup. Let X be a smooth G-space equipped with a G-invariant measure. In this paper, we give upper bounds for K-finite and \mathfrak k{\mathfrak k}-smooth matrix coefficients of the regular representation L ²(X) under an assumption about supp(L²(X)) ?[^(G)]_K{{\rm supp}(L^2(X)) \cap \hat G_K}. Furthermore, we show that this bound holds for unitary representations that are weakly contained in L ²(X). Our result generalizes a result of Cowling–Haagerup–Howe (J Reine Angew Math 387:97–110, 1988). As an example, we discuss the matrix coefficients of the O(p, q) representation L²(\mathbbR^p+q){L^2(\mathbb{R}^{p+q})}.     

On the Converse of the Theory of Groups Acting on Trees with Inversions

In this paper we show that if G is a group acting on a graph X with inversions such that G has a presentation induced by a fundamental domain for the action of G on X, then X is a tree. Received: January 3, 2007., Revised: August 10, 2007 and May 3, 2008., Accepted: October 17, 2008.     

Automorphisms of wonderful varieties

Let G be a complex semisimple linear algebraic group, and X a wonderful G-variety. We determine the connected automorphism group Aut⁰(X) and we calculate Luna’s invariants of X under its action.     

Tournaments with prescribed regular automorphism group

A tournamentX is a TRR for a groupG if (a)G acts regularly on the vertices ofX and (b) Aut(X) is isomorphic toG. We correct some previous work of Babai and Imrich by showing thatZ ₂ ³ andZ ₃ ³ are the only groups of odd order without TRR's. Our methods are perhaps of independent interest, since we use a probabilistic approach.     

Group Actions,k-Derivations,and Finite Morphisms

Let G be an affine algebraic group over an algebraically closed field k of characteristic zero. In this article, we consider finite G-equivariant morphisms F:X → Y of irreducible affine G-varieties. First we determine under which conditions on Y the induced map F ^G :X//G → Y//G of quotient varieties is also finite. This result is reformulated in terms of kernels of derivations on k-algebras A ? B such that B is integral over A. Second we construct explicitly two examples of finite G-equivariant maps F. In the first one, F ^G is quasifinite but not finite. In the second one, F ^G is not even quasifinite.     

Algebraic characterization of isometries of the complex and the quaternionic hyperbolic planes

Let H²_{\mathbb F}{{\bf H}^{\bf 2}_{\mathbb F}} denote the two dimensional hyperbolic space over \mathbb F{\mathbb F} , where \mathbb F{\mathbb F} is either the complex numbers \mathbb C{\mathbb C} or the quaternions \mathbb H{\mathbb H} . It is of interest to characterize algebraically the dynamical types of isometries of H²_{\mathbb F}{{\bf H}^{\bf 2}_{\mathbb F}} . For \mathbb F=\mathbb C{\mathbb F=\mathbb C} , such a characterization is known from the work of Giraud–Goldman. In this paper, we offer an algebraic characterization of isometries of H²_{\mathbb H}{{\bf H}^{\bf 2}_{\mathbb H}} . Our result restricts to the case \mathbb F=\mathbb C{\mathbb F=\mathbb C} and provides another characterization of the isometries of H²_{\mathbb C}{{\bf H}^{\bf 2}_{\mathbb C}} , which is different from the characterization due to Giraud–Goldman. Two elements in a group G are said to be in the same z-class if their centralizers are conjugate in G. The z-classes provide a finite partition of the isometry group. In this paper, we describe the centralizers of isometries of H²_{\mathbb F}{{\bf H}^{\bf 2}_{\mathbb F}} and determine the z-classes.     

Classification of real Nash fibre bundles

Let G be a compact subgroup of an orthogonal group and X an affine, real, semialgebraic Nash variety. A principal Nash G-bundle over X is said to be strongly Nash if it is induced, up to Nash equivalences, of some universal bundle under a Nash map. Not all Nash bundles are strongly Nash and we denote by S(X, G) the class of strongly Nash G-bundles over X. The principal aim of this paper is to prove the following classification theorem: two bundles of S(X, G) are Nash equivalent if and only if they are topologically equivalent; more,there exists a bijection between the family of the classes of Nash equivalent bundles of S(X, G) and , where is the sheaf of germs of the continous maps from X to G. This result leads to find the largest class of principal Nash G-bundles over X in which the topological equivalence always implies the Nash one. Well, we prove that this class is exactly S(X, G). Research partially supported by M.I.U.R.     

Isometries of some Banach spaces of analytic functions

We characterize the surjective isometries of a class of analytic functions on the disk which include the Analytic Besov spaceB _p and the Dirichlet spaceD ^p . In the case ofB _p we are able to determine the form of all linear isometries on this space. The isometries for these spaces are finite rank perturbations of integral operators. This is in contrast with the classical results for the Hardy and Bergman spaces where the isometries are represented as weighted compositions induced by inner functions or automorphisms of the disk.     

Toute variété magnifique est sphérique

LetG be a (connected) reductive group (over C). An algebraicG-varietyX is called “wonderful”, if the following conditions are satisfied:X is (connected) smooth and complete;X containsr irreducible smoothG-invariant divisors having a non void transversal intersection;G has 2 ^r orbits inX. We show that wonderful varieties are necessarily spherical (i.e., they are almost homogeneous under any Borel subgroup ofG).      

The GIT-Equivalence for G-Line Bundles

Let X be a projective variety with an action of a reductive group G. Each ample G-line bundle L on X defines an open subset X^ss(L) of semi-stable points. Following Dolgachev and Hu, define a GIT-class as the set of algebraic equivalence classes of L's with fixed X^ssL. We show that the GIT-classes are the relative interiors of rational polyhedral convex cones, which form a fan in the G-ample cone. We also study the corresponding variations of quotients X^ss(L)//G. This sharpens results of Thaddeus and Dolgachev-Hu.     

Isometric multipliers ofL p (G, X)

Let G be a locally compact group with a fixed right Haar measure andX a separable Banach space. LetL ^p (G, X) be the space of X-valued measurable functions whose norm-functions are in the usualL ^p . A left multiplier ofL ^p (G, X) is a bounded linear operator onB ^p (G, X) which commutes with all left translations. We use the characterization of isometries ofL ^p (G, X) onto itself to characterize the isometric, invertible, left multipliers ofL ^p (G, X) for 1 ≤p ∞,p ≠ 2, under the assumption thatX is not thel ^p -direct sum of two non-zero subspaces. In fact we prove that ifT is an isometric left multiplier ofL ^p (G, X) onto itself then there existsa y ∃ G and an isometryU ofX onto itself such thatTf(x) = U (R _y f)(x). As an application, we determine the isometric left multipliers of L¹ ∩L ^p (G, X) and L¹ ∩C ₀ (G, X) whereG is non-compact andX is not the l^p-direct sum of two non-zero subspaces. If G is a locally compact abelian group andH is a separable Hubert space, we define where г is the dual group of G. We characterize the isometric, invertible, left multipliers ofA ^p (G, H), provided G is non-compact. Finally, we use the characterization of isometries ofC(G, X) for G compact to determine the isometric left multipliers ofC(G, X) providedX ^* is strictly convex.     

On a combinatorial problem in group theory

We say that a groupG ∈DS if for some integerm, all subsetsX ofG of sizem satisfy |X ²|<|X|², whereX ²={xy|x,y ∈X}. It is shown, using a previous result of Peter Neumann, thatG ∈DS if and only if either the subgroup ofG generated by the squares of elements ofG is finite, orG contains a normal abelian subgroup of finite index, on which each element ofG acts by conjugation either as the identity automorphism or as the inverting automorphism. Dedicated to John G. Thompson, the Wolf Prize Laureate in Mathematics for 1992 The first author wishes to thank the Department of Mathematics in the University of Napoli for their hospitality during the preparation of this paper.