In this article we study the space of leftand bi-invariant orderings on a torsion-free nilpotent group G. We will show that generally the set of such orderings is equipped with a faithful action of the automorphism group of G. We prove a result which allows us to establish the same conclusion when G is assumed to be merely residually torsion-free nilpotent. In particular, we obtain faithful act...

Recently, Bismut[ 2] proved a new local index theorem for Dirac operators associated with certain non-torsion-free connections. In this paper, we discuss the odd dimensional analogue. Here the object is the 17-function introduced by Atiyah, Patodi and Singer[l]. And we prove the regularity property of the ~7-function of Dirac operators associated with the appropriate non-torsion-free connection...

Let M be the moduli space of generalized parabolic bundles (GPBs) of rank r and degree d on a smooth curve X . Let ML̄ be the closure of its subset consisting of GPBs with fixed determinant L̄. We define a moduli functor for which ML̄ is the coarse moduli scheme. Using the correspondence between GPBs on X and torsion-free sheaves on a nodal curve Y of which X is a desingularization, we show that M...

Let H be the hyperbolic 3-space and IsomH be the group of isometries of H. Let G be a discrete subgroup of IsomH. The action of G on H extends to a continuous action on the compactification of H by the sphere at infinity S ∞. The limit set Λ(G) is the set of all accumulation points of the orbit of a point in H. The limit set does not depend on the choice of the point in H and the limit set is a...

This paper studies necessary conditions for the existence of α-surfaces in complex space-time manifolds with nonvanishing torsion. For these manifolds, Lie brackets of vector fields and spinor Ricci identities contain explicitly the effects of torsion. This leads to an integrability condition for α-surfaces which does not involve just the self-dual Weyl spinor, as in complexified general relati...

let $a$ be a nite dimensional $k-$algebra and $r$ be a locally bounded category such that $r rightarrow r/g = a$ is a galois covering de ned by the action of a torsion-free group of automorphisms of $r$. following [30], we provide criteria on the convex subcategories of a strongly simply connected category r in order to be a cycle- nite category and describe the module category of $a$. we pro...

We prove that in a torsion-free hyperbolic group, an element is a test element if and only if it is not contained in a proper retract.

Let ΩX be the space of Moore loops on a finite, qconnected, n-dimensional CW complex X , and let R ⊂ Q be a subring containing 1/2. Let ρ(R) be the least non-invertible prime in R. For a graded R-module M of finite type, let FM = M/TorsionM . We show that the inclusion P ⊂ FH∗(ΩX ;R) of the sub-Lie algebra of primitive elements induces an isomorphism of Hopf algebras UP ∼ = −→ FH∗(ΩX ;R), provi...

We show, for each computable ordinal α and degree a > 0(α), the existence of a torsion-free abelian group with proper αth jump degree a.

We construct a finitely presented torsion-free simple group Σ0, acting cocompactly on a product of two regular trees. An infinite family of such groups has been introduced by Burger-Mozes ([2, 4]). We refine their methods and get Σ0 as an index 4 subgroup of a group Σ < Aut(T12)×Aut(T8) presented by 10 generators and 24 short relations. For comparison, the smallest virtually simple group of [4,...