Responses to comments and elaborations of previous posts III

This post is dedicated to the memory of Rabbi Chaim Flom, late rosh yeshiva of Yeshivat Ohr David in Jerusalem. I first met Rabbi Flom thirty years ago when he became my teacher at the Hebrew Youth Academy of Essex County (now known as the Joseph Kushner Hebrew Academy; unfortunately, another one of my teachers from those years also passed away much too young, Rabbi Yaakov Appel). When he first started teaching he was known as Mr. Flom, because he hadn't yet received semikhah (Actually, he had some sort of semikhah but he told me that he didn't think it was adequate to be called "Rabbi" by the students.) He was only at the school a couple of years and then decided to move to Israel to open his yeshiva. I still remember his first parlor meeting which was held at my house. Rabbi Flom was a very special man. Just to give some idea of this, ten years after leaving the United States he was still in touch with many of the students and even attended our weddings. He would always call me when he came to the U.S. and was genuinely interested to hear about my family and what I was working on. He will be greatly missed

Singular surfaces, mod 2 homology, and hyperbolic volume, II

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 g. This improves the main topological result of part I, in which the the same conclusion was obtained under the stronger hypothesis that the dimension of H_1(M;Z_2) is at least 4g-1. As an application we show that if M is a closed orientable hyperbolic 3-manifold with volume at most 3.08, then H_1(M;Z_2) has dimension at most 5.Comment: 23 pages. This version incorporates suggestions from the referee and adds a new section giving examples showing that the main theorem is almost sharp for genus 2. The examples have mod 2 homology of rank 4 and their fundamental groups contain genus 2 surface groups, but they have no closed incompressible surface

Margulis numbers for Haken manifolds

For every closed hyperbolic Haken 3-manifold and, more generally, for any hyperbolic 3-manifold M which is homeomorphic to the interior of a Haken manifold, the number 0.286 is a Margulis number. If M has non-zero first Betti number, or if M is closed and contains a semi-fiber, then 0.292 is a Margulis number for M.Comment: 25 pages. Some statements were clarified some typos were corrected and some of the propositions were generalize