Quasi-isometric classification of non-uniform lattices in semisimple groups of higher rank

We give another proof of the quasi-isometric classification theorem of non-uniform lattices in higher rank semisimple groups. We use the asymptotic cone and a class of ats (the logarithmic ats) which move away in the cusp with logarithmic speed. Submitted: October 1998, Revised version: July 1999, Final version: October 1999.     

Isoperimetric inequalities for nilpotent groups

We prove that every finitely generated nilpotent group of class c admits a polynomial isoperimetric function of degree c + 1 and a linear upper bound on its filling length function.     

On sheets of orbit covers for classical semisimple lie groups

David Vogan gave programmatic conjectures about the Dixmier's map and he made two conjectures that induction may be independent of the choice of parabolic group used and the sheets of orbit data are conjugated or disjointed[1]. In our previous paper, we gave a geometric version of the parabolic induction of the geometric orbit datum (i.e. orbit covers), and proved Vogan's first conjecture for geometric orbit datum:the parabolic induction of the geometric orbit datum is independent of the choice of parabolic group. In this paper, we will prove the other Vogan's conjecture, that is, the sheets are conjugated or disjointed for classical semisimple complex groups.``     

Filling in solvable groups and in lattices in semisimple groups

We prove that the filling order is quadratic for a large class of solvable groups and asymptotically quadratic for all -rank one lattices in semisimple groups of -rank at least 3. As a byproduct of auxiliary results we give a shorter proof of the theorem on the nondistorsion of horospheres providing also an estimate of a nondistorsion constant.     

Irreducible representations of semisimple algebraic groups and supersolvable lattices

Let Λ be the cross section lattice of an irreducible representation of a semisimple algebraic group. Certain combinatorial properties of Λ are studied. Supersolvable Λ?s are determined in terms of Dynkin diagrams. The characteristic polynomial of a supersolvable Λ is computed.     

Isoperimetric inequalities for p-conductance

An n-dimensional domain K is considered with boundary ?K = k₀ tu K₁ ∪ K₂ such that the closure ¯K is the image of a cylinder B=Sx[0, 1] (S is a closed (n?1)-dimensional cell) under a one-one Lipschitz map. For the p-conductance of the domain K, defined by the equation $$c_p (K) = \mathop {\inf }\limits_{U(K)} \int K |\nabla f|^p dx (p > 1),$$ , where∪ (K) = f (x):f ∈ W _p ¹ (K)∩C (¯K),f = 1 on k₁,f = 0 on K₀, the isoperimetric inequality c_p(K) ≤ V/r^P is established. Here V is the n-dimensional volume of the domain K, r is the shortest distance between k₀ and K₁, measured in K. Equality is achieved on the right cylinder.     

The dirac operator and the principal series for complex semisimple lie groups

The Dirac operator plays a fundamental role in the geometric construction of the discrete series for semisimple Lie groups. We show that, at the level of K-theory, the Dirac operator also plays a central role in connection with the principal series for complex connected semisimple Lie groups. This proves the Connes-Kasparov conjecture for such groups.     

Isoperimetric inequalities for sectors in the Minkowski 2-spacetime

In the Minkowski 2-spacetime the hyperbolic angle is defined by the hyperbolic parametrization of the plane. With this notion of hyperbolic angle Helzer obtained a relativistic version of Gauss-Bonnet formula (cf. [3]). In this paper, we derive an isoperimetric inequality for timelike sectors in bounded by an achronal spacelike curve and two timelike rays from a point using this Gauss-Bonnet formula.Supported by grants from TGRC and KRF.     

The word and Riemannian metrics on lattices of semisimple groups

Let G be a semisimple Lie group of rank ⩾2 and Γ an irreducible lattice. Γ has two natural metrics: a metric inherited from a Riemannian metric on the ambient Lie group and a word metric defined with respect to some finite set of generators. Confirming a conjecture of D. Kazhdan (cf. Gromov [Gr2]) we show that these metrics are Lipschitz equivalent. It is shown that a cyclic subgroup of Γ is virtually unipotent if and only if it has exponential growth with respect to the generators of Γ.