Related papers: On volumes of hyperbolic 3-manifolds

We compare the volume of a hyperbolic 3-manifold $M$ of finite volume and the complexity of its fundamental group.

If M is a closed simple 3-manifold whose fundamental group contains a genus-g surface group for some g>1, and if the dimension of H_1(M;Z_2) is at least max(3g-1,6), we show that M contains a closed, incompressible surface of genus at most…

This paper contains a purely topological theorem and a geometric application. The topological theorem states that if M is a simple closed orientable 3-manifold such that \pi_1(M) contains a genus g surface group and H_1(M;Z/2Z) has rank at…

In a variety of settings we provide a method for decomposing a 3-manifold $M$ into pieces. When the pieces have the appropriate type of hyperbolicity, then the manifold $M$ is hyperbolic and its volume is bounded below by the sum of the…

In this paper, we find lower bounds for volumes of hyperbolic 3-manifolds with various topological conditions. Let V_3 = 1.01494 denote the volume of a regular ideal simplex in hyperbolic 3-space. As a special case of the main theorem, if a…

In this paper we derive an explicit lower bound on the volume of a hyperbolic $n$-orbifold for dimensions greater than or equal to four. Our main tool is H. C. Wang's bound on the radius of a ball embedded in the fundamental domain of a…

Periodic geodesics on the modular surface correspond to periodic orbits of the geodesic flow in its unit tangent bundle $\mathrm{PSL}_2(\mathbb{Z})\backslash\mathrm{PSL}_2(\mathbb{R})$. The complement of any finite number of orbits is a…

Let M be a closed hyperbolic 3-manifold. We show that the number of genus g surface subgroups of the fundamental group of M grows like g^{2g}.

In this paper an explicit formula for a lower bound on the volume of a hyperbolic orbifold, dependent on dimension and the maximal order of torsion in the orbifolds' fundamental group, is constructed.

We show that if M is a complete, finite-volume, hyperbolic 3-manifold having exactly one cusp, and if H_1(M;Z_2) has dimension at least 6, then M has volume greater than 5.06. We also show that if M is a closed, orientable hyperbolic…

This note is an elaboration of the ideas and intuitions of Grothendieck and Weil concerning the "arithmetic topology". Given 3-dimensional manifold M fibering over the circle we introduce an real quadratic number field K with discriminant…

In this paper, for each finite group $G$, we construct explicitly a non-compact complete finite-volume arithmetic hyperbolic $4$-manifold $M$ such that $\mathrm{Isom}\,M \cong G$, or $\mathrm{Isom}^{+}\,M \cong G$. In order to do so, we use…

Let ${\mathfrak M}$ be a closed, orientable, hyperbolic 3-orbifold such that $\pi_1({\mathfrak M})$ contains no hyperbolic triangle group. We show that strict upper bounds of 0.07625, 0.1525 and 0.22875 for ${\rm vol}\ {\mathfrak M}$ imply…

We study the problem of bounding the number of cusps of a complex hyperbolic manifold in terms of its volume. Applying algebro-geometric methods using Mumford's work on toroidal compactifications and its generalization due to N. Mok and…

Some conjectures about Heegaard genera and ranks of fundamental groups of 3-manifolds are formulated, and it is shown that they imply new statements about hyperbolic volume.

Let W be a compact manifold and let \rho be a representation of its fundamental group into PSL(2,C). The volume of \rho is defined by taking any \rho-equivariant map from the universal cover of W to H^3 and then by integrating the pull-back…

For a hyperbolic $3$-orbifold with underlying space the $3$-sphere, we obtain a lower bound on its volume in the case that it contains an essential $2$-suborbifold with underlying space the $2$-sphere with four cone points. Our techniques…

We show that if $M$ is any closed, orientable hyperbolic $3$-manifold with ${\rm vol}\ M\le3.69$, we have ${\rm dim}\ H_1(M;{\bf F}_2)\le7$. This may be regarded as a qualitative improvement of a result due to Culler and Shalen, because the…

We give an upper bound for the number of compact essential orientable non-isotopic surfaces, with Euler characteristic at least some constant $\chi$, properly embedded in a finite-volume hyperbolic 3-manifold $M$, closed or cusped. This…

Two different constructions of an invariant of an odd dimensional hyperbolic manifold in the K-group $K_{2n-1}(\bar \Bbb Q)\otimes \Bbb Q$ are given. The volume of the manifold is equal to the value of the Borel regulator on that element.…