2024/03/10 by DeBlois, Jason, Shalen, Peter B.
#57K32 #FOS: Mathematics #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.2403.06058
Let N be a compact, orientable hyperbolic 3-manifold whose boundary is a connected totally geodesic surface of genus 2. If N has Heegaard genus at least 5, then its volume is greater than 2V\rm oct, where V\rm oct=3.66… denotes the volume of a regular ideal hyperbolic octahedron in ℍ3. This improves the lower bound given in our earlier paper ``Volume and topology of bounded and closed hyperbolic 3-manifolds.'' One ingredient in the improved bound is that in a crucial case, instead of using a single ``muffin'' in N in the sense of Kojima and Miyamoto, we use two disjoint muffins. By combining the result about manifolds with geodesic boundary with the log(2k-1) theorem and results due to Agol-Culler-Shalen and Shalen-Wagreich, we show that if M is a closed, orientable hyperbolic 3-manifold with \mathop\rm vol M≤ V\rm oct/2, then dim H1(M;\mathbbF2)≤4. We also provide new lower bounds for the volumes of closed hyperbolic 3-manifolds whose cohomology ring over \mathbbF2 satisfies certain restrictions; these improve results that were proved in ``Volume and topology….''