# Volume and topology of bounded and closed hyperbolic 3-manifolds - Mathematics > Geometric Topology

Volume and topology of bounded and closed hyperbolic 3-manifolds - Mathematics > Geometric Topology - Download this document for free, or read online. Document in PDF available to download.

Abstract: Let N be a compact, orientable hyperbolic 3-manifold with connected, totallygeodesic boundary of genus 2. If N has Heegaard genus at least 5, then itsvolume is greater than 6.89. The proof of this result uses the followingdichotomy: either N has a long return path defined by Kojima-Miyamoto, or Nhas an embedded, codimension-0 submanifold X with incompressible boundary $T\sqcup \partial N$, where T is the frontier of X in N, which is not a book ofI-bundles. As an application of this result, we show that if M is a closed,orientable hyperbolic 3-manifold such that H 1M;Z 2 has dimension at least 5,and if the image in H^2M;Z 2 of the cup product map has image of dimension atmost 1, then M has volume greater than 3.44.

Author: ** Jason DeBlois, Peter B. Shalen**

Source: https://arxiv.org/