Full Isometry Group Research Articles

The skew-torsion holonomy theorem and naturally reductive spaces

We prove a Simons-type holonomy theorem for totally skew 1-forms with values in a Lie algebra of linear isometries. The only transitive case, for this theorem, is the full orthogonal group. We only use geometric methods and we do not use any classification (not even that of transitive isometric actions on the sphere or the list of rank one symmetric spaces). This result was independently proved, by using an algebraic approach, by Paul-Andy Nagy. We apply this theorem to prove that the canonical connection of a compact naturally reductive space is unique, provided the space does not split off, locally, a sphere or a compact Lie group with a bi-invariant metric. From this it follows easily how to obtain the full isometry group of a naturally reductive space. This generalizes known classification results of Onishchick, for normal homogeneous spaces with simple group of isometries, and Shankar, for homogeneous spaces of positive curvature. This also answers a question posed by J. Wolf and Wang-Ziller. Namely, to explain why the presentation group of an isotropy irreducible space, strongly or not, cannot be enlarged (unless for spheres, or for compact simple Lie groups with a bi-invariant metric).

On the [formula omitted] complex reflection group

We give a computer-free proof of a theorem of Basak, describing the group generated by 16 complex reflections of order 3, satisfying the braid and commutation relations of the Y 555 diagram. The group is the full isometry group of a certain lattice of signature ( 13 , 1 ) over the Eisenstein integers Z [ 1 3 ] . Along the way we enumerate the cusps of this lattice and classify the root and Niemeier lattices over Z [ 1 3 ] .

Isometry groups of non-positively curved spaces: structure theory

We develop the structure theory of full isometry groups of locally compact non-positively curved metric spaces. Amongst the discussed themes are de Rham decompositions, normal subgroup structure and characterising properties of symmetric spaces and Bruhat--Tits buildings. Applications to discrete groups and further developments on non-positively curved lattices are exposed in a companion paper: "Isometry groups of non-positively curved spaces: discrete subgroups".

Some properties of non-positively curved lattices

We announce results on the structure of CAT(0) groups, CAT(0) lattices and of the underlying spaces. Our statements rely notably on a general study of the full isometry groups of proper CAT(0) spaces. Classical statements about Hadamard manifolds are established for singular spaces; new arithmeticity and rigidity statements are obtained. To cite this article: P.-E. Caprace, N. Monod, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

Boundaries of cocompact proper CAT(0) spaces

A proper CAT(0) metric space X is cocompact if it has a compact generating domain with respect to its full isometry group. Any proper CAT(0) space, cocompact or not, has a compact metrizable boundary at infinity ∂ ∞ X ; indeed, up to homeomorphism, this boundary is arbitrary. However, cocompactness imposes restrictions on what the boundary can be. Swenson showed that the boundary of a cocompact X has to be finite-dimensional. Here we show more: the dimension of ∂ ∞ X has to be equal to the global Čech cohomological dimension of ∂ ∞ X . For example: a compact manifold with non-empty boundary cannot be ∂ ∞ X with X cocompact. We include two consequences of this topological/geometric fact: (1) The dimension of the boundary is a quasi-isometry invariant of CAT(0) groups. (2) Geodesic segments in a cocompact X can “almost” be extended to geodesic rays, i.e. X is almost geodesically complete.

Some isometry groups of the Urysohn space

We construct various isometry groups of the Urysohn space (the unique complete separable metric space that is universal and homogeneous), including abelian groups which act transitively, and free groups which are dense in the full isometry group.

Symmetries of Eschenburg spaces and the Chern problem

To advance our basic knowledge of manifolds with positive (sectional) curvature it is essential to search for new examples, and to get a deeper understanding of the known ones. Although any positively curved manifold can be perturbed so as to have trivial isometry group, it is natural to look for, and understand the most symmetric ones, as in the case of homogeneous spaces. In addition to the compact rank one symmetric spaces, the complete list (see [BB]) of simply connected homogeneous manifolds of positive curvature consists of the Berger spaces B7 and B13 [Be], the Wallach spaces W 6,W 12 and W 24 [Wa], and the infinite class of socalled Aloff–Wallach spaces, A7 [AW]. Their full isometry groups were determined in [Sh2], and this knowledge provided new basic information about possible fundamental groups of positively curved manifolds, and in particular to counter-examples of the so-called Chern conjecture (see [Sh1] and [GSh, Ba2]), which states that every abelian subgroup of the fundamental group is cyclic. Our purpose here is to begin a systematic analysis of the isometry groups of the remaining known manifolds of positive curvature, i.e., of the so-called Eschenburg spaces, E7 [Es1, Es2] (plus one in dimension 6) and the Bazaikin spaces, B13 [Ba1], with an emphasis on the former. In particular, we completely determine the identity component of the isometry group of any positively curved Eschenburg space. A member of E is a so-called bi-quotient of SU(3) by a circle:

Finite groups and hyperbolic manifolds

The isometry group of a compact n-dimensional hyperbolic manifold is known to be finite. We show that for every n≥2, every finite group is realized as the full isometry group of some compact hyperbolic n-manifold. The cases n=2 and n=3 have been proven by Greenberg (1974) and Kojima (1988), respectively. Our proof is non constructive: it uses counting results from subgroup growth theory to show that such manifolds exist.

Weak Symmetry in Naturally Reductive Homogeneous Nilmanifolds

We prove that within the class of naturally reductive homogeneous nilmanifolds, the notions of weak symmetry, i.e., any two points can be interchanged by an isometry, and commutativity, i.e., isometry invariant differential operators commute, are equivalent. A connected Riemannian manifold M is said to be weakly symmetric if for any two points p, q ∈ M there exists an isometry of M mapping p to q and q to p. These spaces were introduced by Selberg in the framework of his development of the trace formula, see [12], where it is proved that in a weakly symmetric space M , the algebra of all invariant (with respect to the full isometry group I(M)) differential operators on M is commutative, that is, M is a commutative space. Selberg asks in [12] whether the converse holds. The answer is negative, the known counterexamples arise in certain homogeneous nilmanifolds, so-called H-type groups, see [8, 9]. On the other hand, it has been proved by Akhiezer and Vinberg [1] that, for homogeneous spaces of reductive algebraic groups, the answer is affirmative. In such a way, a natural question takes place: under what extra conditions on the homogeneous Riemannian manifold M , the weak symmetry is necessary for the commutativity of I(M)-invariant differential operators? With such a question in mind, the first observation we make is that none of the counterexamples found is naturally reductive. A Riemannian manifold M is said to be naturally reductive, if there exists a transitive Lie group G of isometries with isotropy subgroup K at p ∈ M , and an Ad (K)-invariant vector subspace m of g complementary 2000 AMS Mathematics Subject Classification. Primary 22E30, 53C30, Secondary 22E25, 43A20.

ISOMETRIES OF ELLIPTIC 3-MANIFOLDS

The closed 3-manifolds of constant positive curvature were classified long ago by Seifert and Threlfall. Using well-known information about the orthogonal group O(4), we calculate their full isometry groups Isom(M), determine which elliptic 3-manifolds admit Seifert fiberings that are invariant under all isometries, and verify that the inclusion of Isom(M) to Diff(M) is a bijection on path components.

Isometry groups of homogeneous spaces with positive sectional curvature

We calculate the full isometry group in the case G/ H admits a homogeneous metric of positive sectional curvature.

Isometry groups of homogeneous quaternionic Kähler manifolds

A general method for calculation of the full isometry group of a Riemannian solvmanifold is presented.

Commutative Spaces Which are Not Weakly Symmetric

In 1956, A. Selberg introduced weakly symmetric spaces in the framework of his development of the trace formula, and proved that in a weakly symmetric space, the algebra of all invariant (with respect to the full isometry group) differential operators is commutative [ 22 ]. In this paper, Selberg asked whether the converse holds. In the present work, we shall answer this question by presenting examples of commutative spaces which are not weakly symmetric. These examples arise in the quaternionic analogues to the Heisenberg group, endowed with certain special metrics (see Theorem 5 and the explicit realization after it).

Pólya theory of hypercubes (revision of preprint 925)

The cycle index polynomial is described for the action of the full isometry group of the n-dimensional hypercube on its q-dimensional cells. This group action is interpreted as \(C_2^n \cdot S_n \) acting on the set of unit basis vectors in Rn and their opposites. A kind of generating function that yields all these polynomials at once is obtained by Mobius inversion. The same technique is applied to the simpler case of the n-dimensional simplex.

Isometry transformations of hyperbolic 3-manifolds

The isometry group of a compact hyperbolic manifold is known to be finite. We show that every finite group is realized as the full isometry group of some compact hyperbolic 3-manifold.

The fine structure of transitive Riemannian isometry groups. I

Let M M be a connected homogeneous Riemannian manifold, G G the identity component of the full isometry group of M M and H H a transitive connected subgroup of G G . G = H L G = HL , where L L is the isotropy group at some point of M M . M M is naturally identified with the homogeneous space H / H ∩ L H/H \cap L endowed with a suitable left-invariant Riemannian metric. This paper addresses the problem: Given a realization of M M as a Riemannian homogeneous space of a connected Lie group H H , describe the structure of the full connected isometry group G G in terms of H H . This problem has already been studied in case H H is compact, semisimple of noncompact type, or solvable. We use the fact that every Lie group is a product of subgroups of these three types in order to study the general case.

Finite groups as isometry groups

We show that given any finite group G of cardinality k + 1 k + 1 , there is a Riemannian sphere S k − 1 {S^{k - 1}} (imbeddable isometrically as a hypersurface in R k {{\mathbf {R}}^k} ) such that its full isometry group is isomorphic to G. We also show the existence of a finite metric space of cardinality k ( k + 1 ) k(k + 1) whose full isometry group is isomorphic to G.