Three-dimensional hyperelliptic manifolds

Let X³ = H³, E³, S³, H² × E¹, S² × E¹, T₁(H²), Nil of Solv be one of the eight 3-dimensional geometrics of Thurston [10] and G be a discrete group of isometrics of X³ acting without fixed points. A manifold M³ = X³/G is said to be hyperelliptic if there is an isometric involution on it such that the factor space M³/<> is diffeomorphic to the 3-sphere S³. In analogy with the theory of Riemann surfaces we call involution.In the present paper the existence of hyperelliptic manifolds in each light of the eight 3-dimensional geometrics will be obtained. All the proofs given there will be written in the language of orbifolds whose basic facts can be found in [9].     

Systolic volume of hyperbolic manifolds and connected sums of manifolds

The systolic volume of a closed n-manifold M is defined as the optimal constant σ(M) satisfying the inequality vol(M, g) ≥ σ(M) sys(M, g) ⁿ between the volume and the systole of every metric g on M. First, we show that the systolic volume of connected sums of closed oriented essential manifolds is unbounded. Then, we prove that the systolic volume of every sequence of closed hyperbolic (three-dimensional) manifolds is also unbounded. These results generalize systolic inequalities on surfaces in two different directions.      

Moduli spaces over manifolds with involutions

Most of the work was carried out under the support of K.C. Wong Education Foundation Ltd.     

Clifford algebras of hyperbolic involutions

For a central simple algebra of even degree with hyperbolic orthogonal involution, we describe the canonically induced involution on the even Clifford algebra of . When , and the interesting part of is isomorphic to the canonical involution on an exterior power algebra of B. As a corollary, we get some properties of the involution on the exterior power algebra. Received January 27, 1998; in final form June 7, 2000 / Published online October 30, 2000     

2-Weierstrass points of genus 3 hyperelliptic curves with extra involutions

We consider families of curves with extra automorphisms in ?₃, the moduli space of smooth hyperelliptic curves of genus g = 3. Such families of curves are explicitly determined in terms of the absolute invariants of binary octavics. For each family of positive dimension where |Aut (C)|>4, we determine the possible distributions of weights of 2-Weierstrass points.     

Analytic torsion of complete hyperbolic manifolds of finite volume

In this paper we define the analytic torsion for a complete oriented hyperbolic manifold of finite volume. It depends on a representation of the fundamental group. For manifolds of odd dimension, we study the asymptotic behavior of the analytic torsion with respect to certain sequences of representations obtained by restriction of irreducible representations of the group of isometries of the hyperbolic space to the fundamental group.     

Three-dimensional manifolds of nonpositive curvature with small injectivity radius

It is proved that if at every point of a closed, three-dimensional, Riemannian manifold with bounded sectional curvature the injectivity radius does not exceed a specific absolute constant, then the manifold is a special graph and its metric splits locally.Translated from Ukrainskii Geometricheskii Sbornik, No. 34, pp. 10–19, 1991.     

Compact hyperbolic 4-manifolds of small volume

We prove the existence of a compact non-orientable hyperbolic 4-manifold of volume and a compact orientable hyperbolic 4-manifold of volume , obtainable from torsion-free subgroups of small index in the Coxeter group . At the time of writing these are the smallest volumes of any known compact hyperbolic 4-manifolds.

     

On involutions with middle-dimensional fixed-point locus and holomorphic-symplectic manifolds

Let $g$ be an involution of a compact closed manifold $X$ such that the fixed-point set $X^{g}$ is middle dimensional. Under the assumption that the normal bundle of the fixed-point set is either the tangent or co-tangent bundle conditions on the equivariant invariants of $X$ arise. In particular if $X$ is a holomorphic-symplectic manifold and $g$ an anti holomorphic-symplectic involution one arrives at a generalisation of Beauville’s result that for $X$ a hyper-Kähler manifold the $\hat{A}$ genus of $X^{g}$ is one.     

Notes on the peripheral volume of hyperbolic 3‐manifolds

We consider orientable hyperbolic 3‐manifolds with either non‐empty compact geodesic boundary, or some toric cusps, or both. For any such M we analyze what portion of the volume of M can be recovered by inserting in M boundary collars and cusp neighbourhoods with disjoint embedded interiors. Our main result is that this portion can only be maximal in some combinatorially extremal configurations. The techniques we employ are very elementary but the result is in our opinion of some interest.