Parameswaran Sankaran

Twisted conjugacy classes in abelian extensions of certain linear groups

Given a group automorphism o:Γ→Γ, one has an action of Γ on itself by o-twisted conjugacy, namely,g.x=gxog -1 ). The orbits of this action are called o-twisted conjugacy classes. One says that has the R∞-property if there are infinitely many o-twisted conjugacy classes for every automorphism o of Γ. In this paper we show that SL(n,Z) and its congruence subgroups have the R∞-property. Further we show that any (countable) abelian extension of Γ has the R∞-property where Γ is a torsion free non-elementary hyperbolic group, or SL(n,Z),Sp,(2n,Z) or a principal congruence subgroup of SL(n,Z) or the fundamental group of a complete Riemannian manifold of constant negative curvature.

Chaotic group actions on manifolds

Abstract We construct chaotic actions of certain finitely generated infinite abelian groups on even-dimensional spheres, and of finite index subgroups of SL n ( Z ) on tori. We also study chaotic group actions via compactly supported homeomorphisms on open manifolds.

A new approach to standard monomial theory for classical groups

We describe a procedure for constructing monomial bases for finite dimensional irreducible representations of complex semisimple Lie algebras. A basis is calledmonomial if each of its elements is the result of applying to a (fixed) highest weight vector a monomial in the Chevalley basis elementsYα, α a simple root, in the opposite Borel subalgebra. As an immediate application we obtain a new proof of the main theorem of standard monomial theory for classical groups.

TWISTED CONJUGACY CLASSES IN LATTICES IN SEMISIMPLE LIE GROUPS

Given a group automorphism ϕ: Γ → Γ, one has an action of Γ on itself by ϕ-twisted conjugacy, namely, g.x = gxϕ(g−1). The orbits of this action are called ϕ-conjugacy classes. One says that Γ has the R∞-property if there are infinitely many ϕ-conjugacy classes for every automorphism ϕ of Γ. In this paper we show that any irreducible lattice in a connected semisimple Lie group having finite centre and rank at least 2 has the R∞-property.

Stable parallelizability of partially oriented flag manifolds II

In the first paper with the same title the authors were able to determine all partially oriented flag manifolds that are stably parall elizable or parallelizable, apart from four infinite families that were undecided. Here, using more delicate techniques (mainly K-theory), we settle these previously undecided families and show that none of the manifolds in them is stably parallelizable, apart from o ne 30-dimensional manifold which still remains undecided.

On degrees of maps between Grassmannians

Let Gn,k denote the oriented grassmann manifold of orientedk-planes in ℝn. It is shown that for any continuous mapf: Gn,k → Gn,k, dim Gn,k = dim Gm,l = l(m −l), the Brouwer’s degree is zero, providedl > 1,n ≠ m. Similar results for continuous mapsg: ℂGm,l → ℂGn,k,h: ℍGm,l → ℍGn,k, 1 ≤ l < k ≤ n/2, k(n — k) = l(m — l) are also obtained.

On continuous maps between Grassmann manifolds

LetGn,k denote the Grassmann manifold ofk-planes in ℝn. We show that for any continuous mapf: Gn,k→Gn,l the induced map inZ/2-cohomology is either zero in positive dimensions or has image in the subring generated by w1(γn, k), provided 1≤l<k≤[n/2] andn≥k+2l-1. Our main application is to obtain negative results on the existence of equivariant maps between oriented Grassmann manifolds. We also obtain positive results in many cases on the existence of equivariant maps between oriented Grassmann manifolds.

Determination of Grassmann manifolds which are boundaries

Let FG nk denote the Grassmann manifold of all k-dimensional (left) F-vector subspace of F n for F = R, the reals, C, the complex numbers, or H the quaternions. The problem of determining which of the Grassmannians bound was addressed by the author in [4]. Partial results were obtained in [4] for the case F = R, including a sufficient condition, due to A. Dold, on n and k for R G nk to bound. Here, we show that Dolds condition is also necessary, and obtain a new proof of sufficiency using the methods of this paper, which cover the complex and quaternionic cases as well.

A coincidence theorem for holomorphic maps to G/P

The purpose of this note is to extend to an arbitrary generalized Hopf and Calabi- Eckmann manifold the following result of Kalyan Mukherjea: Let Vn = S 2n+1 × S 2n+1 denote a Calabi-Eckmann manifold. If f,g : Vn−→P n are any two holomorphic maps, at least one of them being non-constant, then there exists a coincidence: f(x) = g(x) for some x ∈ Vn. Our proof involves a coincidence theorem for holomorphic maps to complex projective varieties of the form G/P where G is complex simple algebraic group and P ⊂ G is a maximal parabolic subgroup, where one of the maps is dominant.

Elementary abelian 2-group actions on flag manifolds and applications

Let N ∗ denote the unoriented cobordism ring. Let G=(Z=2) n and let Z ∗ (G) denote the equivariant cobordism ring of smooth manifolds with smooth G-actions having finite stationary points. In this paper we show that the unoriented cobordism class of the (real) flag manifold M=O(m)=(O(m 1 )× . . . ×O(m s )) is in the subalgebra generated by ⊕ i n N i , where m=Σ m j , and 2 n−1 n . We obtain sufficient conditions for in-decomposability of an element in Z ∗ (G). We also obtain a sufficient condition for algebraic independence of any set of elements in Z ∗ (G). Using our criteria, we construct many indecomposable elements in the kernel of the forgetful map Z d (G)→N d in dimensions 2 ≤ d 2, and show that they generate a polynomial subalgebra of Z * (G).