Classification of torus manifolds with codimension one extended actions

The purpose of this paper is to classify torus manifolds (M ²ⁿ , T ⁿ ) with codimension one extended G-actions (M ²ⁿ , G) up to essential isomorphism, where G is a compact, connected Lie group whose maximal torus is T ⁿ . For technical reasons, we do not assume torus manifolds are orientable. We prove that there are seven types of such manifolds. As a corollary, if a nonsingular toric variety or a quasitoric manifold has a codimension one extended action then such manifold is a complex projective bundle over a product of complex projective spaces.     

Stability of compact actions of ℝ n of codimensional oneof codimensional one

In this paper we study theC ^r stability of locally free compact actions of ℝ ⁿ with compact orbits over manifolds of dimensionn+1. More precisely, we show that in many cases aC ¹ perturbation of an action with all orbits compact must also have all orbits compact and aC ⁰ perturbation usually has many compact orbits.     

On space forms of Grassmann manifolds

We show that, in each odd dimensionn =m ², there is a class of Grassmann quotient spaces not included in Wolf’s classic solution of the Grassmann space form problem. We classify all of these new Grassmann space forms up to isometry. As an application, we exhibit a pair of compact Einstein manifolds of dimensionm ² with holonomy groups which are abstractly isomorphic yet not conjugate in the orthogonal group, thus proving that a theorem of Besse cannot be extended to non-simply-connected Einstein manifolds.     

The Socle and finite dimensionality of some Banach algebras

The purpose of this note is to describe some algebraic conditions on a Banach algebra which force it to be finite dimensional. One of the main results in Theorem 2 which states that for a locally compact groupG, G is compact if there exists a measure μ in Soc(L ¹(G)) such that μ(G) ≠ 0. We also prove thatG is finite if Soc(M(G)) is closed and every nonzero left ideal inM(G) contains a minimal left ideal.     

Cohomology of compact hyperkähler manifolds and its applications

We announce the structure theorem for theH ²(M)-generated part of cohomology of a compact hyperkähler manifold. This computation uses an action of the Lie algebra so(4,n–2) wheren=dimH ²(M) on the total cohomology space ofM. We also prove that every two points of the connected component of the moduli space of holomorphically symplectic manifolds can be connected with so-called twistor lines — projective lines holomorphically embedded in the moduli space and corresponding to the hyperkähler structures. This has interesting implications for the geometry of compact hyperkähler manifolds and of holomorphic vector bundles over such manifolds.     

The geodesic flow of a nonpositively curved graph manifold

We consider discrete cocompact isometric actions where X is a locally compact Hadamard space (following [B] we will refer to CAT(0) spaces — complete, simply connected length spaces with nonpositive curvature in the sense of Alexandrov — as Hadamard spaces) and G belongs to a class of groups (“admissible groups”) which includes fundamental groups of 3-dimensional graph manifolds. We identify invariants (“geometric data”) of the action which determine, and are determined by, the equivariant homeomorphism type of the action of G on the ideal boundary of X. Moreover, if are two actions with the same geometric data and is a G-equivariant quasi-isometry, then for every geodesic ray there is a geodesic ray (unique up to equivalence) so that . This work was inspired by (and answers) a question of Gromov in [Gr3, p. 136]. Submitted: May 2001.     

Prime ideals in the C*-algebra of a nilpotent group

We say that a locally compact groupG hasT ₁ primitive ideal space if the groupC ^*-algebra,C ^*(G), has the property that every primitive ideal (i.e. kernel of an irreducible representation) is closed in the hull-kernel topology on the space of primitive ideals ofC ^*(G), denoted by PrimG. This means of course that every primitive ideal inC ^*(G) is maximal. Long agoDixmier proved that every connected nilpotent Lie group hasT ₁ primitive ideal space. More recentlyPoguntke showed that discrete nilpotent groups haveT ₁ primitive ideal space and a few month agoCarey andMoran proved the same property for second countable locally compact groups having a compactly generated open normal subgroup. In this note we combine the methods used in [3] with some ideas in [9] and show that for nilpotent locally compact groupsG, having a compactly generated open normal subgroup, closed prime ideals inC ^*(G) are always maximal which implies of course that PrimG isT ₁.     

Strong Lefschetz property under reduction

Let n > 1 and G be the group SU(n) or Sp(n). This paper constructs compact symplectic manifolds whose symplectic quotient under a Hamiltonian G-action does not inherit the strong Lefschetz property.     

Spectral synthesis in regular Banach algebras

We prove that every locally compact non-discrete abelian groupG contains a compact subsetE such thatA(E) — the restriction algebra ofA(G) toE — admits spectral synthesis, although it contains a closed, regular, self-adjoint subalgebra which is isomorphic to an algebra of infinitely differentiable functions on [−1, 1]. We also give some general results concerning the failure of spectral synthesis in regular Banach algebras. This paper is a part of the author’s Ph.D. thesis prepared at the Hebrew University of Jerusalem, under the supervision of Professor Y. Katznelson, to whom the author wishes to express his gratitude for his valuable remarks, and the interest he showed in the preparation of this paper.     

Irreducible decompositions of unitary representations of infinite- dimensional groups

This paper concerns the problem of irreducible decompositions of unitary representations of topological groups G, including the group Diff₀(M) of diffeomorphisms with compact support on smooth manifolds M. It is well known that the problem is affirmative, when G is a locally compact, separable group (cf. [3, 4]). We extend this result to infinite-dimensional groups with appropriate quasi-invariant measures, and, in particular, we show that every continuous unitary representation of Diff₀(M) has an irreducible decomposition under a fairly mild condition. This research was partially supported by a Grant-in-Aid for Scientific Research (No.14540167), Japan Socieity of the Promotion of Science.     

Elliptic Symbols

If G is the structure group of a manifold M it is shown how a certain ideal in the character ring of G corresponds to the set of geometric elliptic operators on M. This provides a simple method to construct these operators. For classical structure groups like G = O(n) (Riemannian manifolds), G = SO(n) (oriented Riemannian manifolds), G = U(m) (almost complex manifolds), G = Spin(n) (spin manifolds), or G = Spin^c(n) (spin^c manifolds) this yields well known classical operators like the Euler—deRham operator, signature operator, Cauchy—Riemann operator, or the Dirac operator. For some less well studied structure groups like Spin^h(n) or Sp(q)Sp(1) we can determine the corresponding operators. As applications, we obtain integrality results for such manifolds by applying the Atiyah—Singer Index Theorem to these operators. Finally, we explain how immersions yield interesting structure groups to which one can apply this method. This yields lower bounds on the codimension of immersions in terms of topological data of the manifolds involved.     

On minimal,strongly proximal actions of locally compact groups

Minimal, strongly proximal actions of locally compact groups on compact spaces, also known asboundary actions, were introduced by Furstenberg in the study of Lie groups. In particular, the action of a semi-simple real Lie groupG on homogeneous spacesG/Q, whereQ ⊂G is a parabolic subgroup, are boundary actions. Countable discrete groups admit a wide variety of boundary actions. In this note we show that ifX is a compact manifold with a faithful boundary action of some locally compact groupH, then (under some mild regularity assumption) the groupH, the spaceX, and the action split into a direct product of a semi-simple Lie groupG acting onG/Q and a boundary action of a discrete countable group. The author was partially supported by NSF grants DMS-0049069, 0094245 and GIF grant G-454-213.06/95.     

Examples of amalgamated products

The rank of a groupG is the minimal number of elements that generateG. For any natural numbern we construct two groups,G ₁ of rankr(G ₁)=n andG ₂ of rankr(G ₂)=2n such that their amalgamated product over an infinite cyclic subgroup, malnormal in both factors, is generated by 2n=r(G ₁)+r(G ₂)−n elements. We also consider an example of an amalgamated product ofn factors: such thatr(G)=n +1, andr(A)≥1. This example realizes the lower bound given by Weidmann [W1] (see Theorem 2 in the present paper).     

Distal actions and shifted convolution property

A locally compact group G is said to have shifted convolution property (abbr. as SCP) if for every regular Borel probability measure μ on G, either sup _x∈G μ ⁿ (Cx) → 0 for all compact subsets C of G, or there exist x ∈ G and a compact subgroup K normalised by x such that μ ⁿ x ⁻ⁿ → ωK, the normalised Haar measure on K. We first consider distality of factor actions of distal actions. It is shown that this holds in particular for factors by compact groups invariant under the action and for factors by the connected component of the identity. We then characterize groups having SCP in terms of a readily verifiable condition on the conjugation action (pointwise distality). This gives some interesting corollaries to distality of certain actions and Choquet-Deny measures which actually motivated SCP and pointwise distal groups. We also relate distality of actions on groups to that of the extensions on the space of probability measures.     

Polynomial mappings of groups

A mapping ϕ of a groupG to a groupF is said to be polynomial if it trivializes after several consecutive applications of operatorsD _h ,h ∈G, defined byD _h ϕ(g)=ϕ(g) ⁻¹ ϕ(gh). We study polynomial mappings of groups, mainly to nilpotent groups. In particular, we prove that polynomial mappings to a nilpotent group form a group with respect to the elementwise multiplication, and that any polynomial mappingG→F to a nilpotent groupF splits into a homomorphismG→G’ to a nilpotent groupG’ and a polynomial mappingG’→F. We apply the obtained results to prove the existence of the compact/weak mixing decomposition of a Hilbert space under a unitary polynomial action of a finitely generated nilpotent group. This work was supported by NSF, Grants DMS-9706057 and 0070566.     

The Embedding W n,1 into C 0 on Compact Riemannian Manifolds

We study the best constant in the inequality corresponding to the Sobolev embedding W ^n,1(R ⁿ ) into the space of bounded continuous functions C ₀(R ⁿ ). Then, we adapt this inequality on compact Riemannian manifolds and discuss on its optimality.     

On the commutativity degree of compact groups

In any finite group G, the commutativity degree of G (denoted by d(G)) is the probability that two randomly chosen elements of G commute. More generally, for every n ≥ 2 the nth commutativity degree (denoted by d _n (G)) is the probability that a randomly chosen ordered (n + 1)-tuple of the group elements is mutually commuting. The aim of this paper is to generalize the definition of d(G) and d _n (G) to every compact group G (infinite and even uncountable). We shall state some results concerning compact groups and we will extend some results in Erfanian et al. (Comm. Algebra 35 (2007), 4183–4197) and Lescot (J. Algebra 177 (1995), 847–869).     

Almost constant sequences and well-distribution in compact groups

We study the properties of sequences (c _n ) in a compact groupG such that (x _n ) being (weakly) well-distributed implies (c _n x _n ) being (weakly) well-distributed and give a complete characterisation in terms of almost constant sequences as well as a generalisation to non-compact locally compact groups.     

Automorphism groups of graphs with forbidden subgraphs

For a graph Ф letF(Ф) be the class of finite graphs which do not contain an induced subgraph isomorphic to Ф. We show that whenever Ф is not isomorphic to a path on at most 4 vertices or to the complement of such a graph then for every finite groupG there exists a graph ГєF(Ф) such thatG is isomorphic to the automorphism group of Г. For all paths д on at most 4 vertices we determine the class of all automorphism groups of members ofF(д).     

Compact generalized Hopf and cosymplectic solvmanifolds and the Heisenberg groupH(n, 1)

In this paper we obtain a generalized Hopf structure on the total space of certain principal circle bundles over a compact cosympletic manifold. Using this result we give new examples of compact generalized Hopf manifolds. We describe these examples as suspensions with fibre a compact quotient of the generalized Heisenberg groupH(n, 1) by a discrete subgroup and we show an explicit realization of them as compact solvmanifolds.