Final Report: 1122020

Final Report for Period: 09/2010 - 08/2011 Submitted on: 11/29/2011 Principal Investigator: Margalit, Dan . Award ID: 1122020 Organization: Georgia Tech Research Corp Submitted By: Margalit, Dan - Principal Investigator Title: Algebra and of the Johnson filtration

Project Participants Senior Personnel Name: Margalit, Dan Worked for more than 160 Hours: Yes Contribution to Project:

Post-doc

Graduate Student

Undergraduate Student

Technician, Programmer

Other Participant

Research Experience for Undergraduates

Organizational Partners

Other Collaborators or Contacts Mladen Bestvina, University of Utah Kai-Uwe Bux, Bielefeld University Tara Brendle, University of Glasgow Benson Farb, University of Chicago Chris Leininger, University of Illinois at Urbana-Champaign Saul Schleimer, University of Warwick Andy Putman, Rice University Leah Childers, Pittsburg State University Allen Hatcher,

Activities and Findings

Research and Education Activities: Proved various theorems, listed below. Wrote papers explaining the results. Wrote involved computer program to help explore the quotient of Outer space by the Torelli group for Out(F_n). Gave lectures on the subject of this Project at a conference for graduate students called 'Examples of Groups' at Ohio State University, June

Page 1 of 5 Final Report: 1122020 2008. Mentored graduate students at the University of Utah in the subject of mapping class groups. Interacted with many students about the book I am writing with Benson Farb on mapping class groups. Gave many talks on this work. Taught a graduate course on my research area, using a book that I translated with Djun Kim. Organized a special session on Geometric Group Theory at MathFest. Finished book with Benson Farb, published by Press. Currently finishing a book with Djun Kim, to be published at Princeton University Press. In early stages of a book on Geometric Group Theory, edited with Matt Clay. Advising three graduate students, Rebecca Winarski, Marta Aguilera (visiting GaTech from Sevilla), and Hyunshik Shin. Winarksi is finishing her first paper, and Shin has made exciting progress on his research project. Organizing Topology Students Workshop for Summer 2012. Organizing Tech Topology Conference for December 2011. Running reading course with ~15 participants on Geometric Group Theory.

Findings: Computed cohomological dimension of the first two terms of the Johnson filtration, and gave narrow bounds for each other term. Found combinatorial proof that the genus 2 Torelli group is an infinitely generated free group. Constructed a new contractible complex on which the Torelli group acts, and gave two proofs of contractibility. Classified multitwists in each term of the Johnson filtration. Computed the cohomological dimension of the hyperelliptic Torelli group, and showed that the top homology group is infinitely generated. Found first examples of nontrivial roots of Dehn twists about nonseparating curves. Proved that any two elements of the pure braid group either commute or generate a free group. Showed that only finitely many 3-manifolds appear as mapping tori of low dilatation pseudo-Anosovs, up to Dehn surgery. Proved a Birman exact sequence for the hyperelliptic Torelli group. Made progress towards proving that hyperelliptic Torelli groups are generated by reducible elements. In particular gave al algebraic description of the quotient of the complex of symmetric nonseparating curves. Gave explicit factorization of certain basic elements of the hyperelliptic Torelli group into Dehn twists. Described all torsion elements of the level 2 . Described the failure of this group to be pure. Gave a simple proof that the Torelli group is generated by bounding pair maps.

Page 2 of 5 Final Report: 1122020 Described the location of short geodesics in moduli space - they live in the uniformly thick/thin part. Showed that the number of short geodesics in moduli space is bounded from above and below by polynomials.

Training and Development: I have sought the advice and help from a number of graduate students about the book that I am writing with Benson Farb. This has been a valuable experience for these students, as it requires them to take an active role in the advanced mathematics they are learning. Taught an undergraduate REU class and was a teaching mentor to the teaching assistant, Utah graduate student William Malone. By working with Bestvina, Bux, Brendle, Schleimer, Leininger, and Farb on the above mentioned theorems, I gained a great deal of research experience. Worked with graduate students during a topics course. Advised final projects and organized mini-conference with their presentations. Instituted online homework for calculus courses at Tufts. Taught an intensive undergraduate course in knot theory. Produced notes that have been disseminated to others teaching similar courses. Taught a graduate course in . Currently advising three graduate students. Running reading course on Geometric Group Theory with ~15 students. Organizing Tech Topology Conference, December 2011. Organizing Topology Students Workshop for Summer 2011.

Outreach Activities: I helped to organize the Calculus Carnival at the University of Utah, which was an attempt to make Calculus more appealing to undergraduates. A writeup of this event was published in FOCUS magazine, published by the MAA. I taught a minicourse at Ohio State University on the subject of this project. Worked with graduate students in my topics course. Answered many questions, mostly via email, about both book projects. Organized a session on Geometric Group Theory at Mathfest August 2010. Plenary lecture at Young Mathematicians Conference at OSU. Lectured to undergraduate math majors at GaTech. Instituted departmental tea at GaTech.

Journal Publications

Bestvina, M; Bux, KU; Margalit, D, "Dimension of the Torelli group for Out(F-n)", INVENTIONES MATHEMATICAE, p. 1, vol. 170, (2007). Published, 10.1007/s00222-007-0055-

Bell, RW; Margalit, D, "Injections of Artin groups", COMMENTARII MATHEMATICI HELVETICI, p. 725, vol. 82, (2007). Published,

Page 3 of 5 Final Report: 1122020 Farb, B; Leininger, CJ; Margalit, D, "The lower central series and pseudo-Anosov dilatations", AMERICAN JOURNAL OF MATHEMATICS, p. 799, vol. 130, (2008). Published,

Brendle, TE; Margalit, D, "Commensurations of the Johnson kernel (vol 8, pg 1361, 2004)", GEOMETRY & TOPOLOGY, p. 97, vol. 12, (2008). Published, 10.2140/gt.2008.12.9

Behrstock, J; Margalit, D, "Curve complexes and finite index subgroups of mapping class groups", GEOMETRIAE DEDICATA, p. 71, vol. 118, (2006). Published, 10.1007/s10711-005-9022-

Margalit, D; Spallone, S, "A homological recipe for pseudo-Anosovs", MATHEMATICAL RESEARCH LETTERS, p. 853, vol. 14, (2007). Published,

Leininger, CJ; Margalit, D, "Two-generator subgroups of the pure braid group", GEOMETRIAE DEDICATA, p. 107, vol. 147, (2010). Published, 10.1007/s10711-009-9440-

Bestvina, M; Bux, KU; Margalit, D, "THE DIMENSION OF THE TORELLI GROUP", JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY, p. 61, vol. 23, (2010). Published,

Margalit, D; Schleimer, S, "Dehn twists have roots", GEOMETRY & TOPOLOGY, p. 1495, vol. 13, (2009). Published, 10.2140/gt.2009.13.149

Margalit, D; McCammond, J, "GEOMETRIC PRESENTATIONS FOR THE PURE BRAID GROUP", JOURNAL OF KNOT THEORY AND ITS RAMIFICATIONS, p. 1, vol. 18, (2009). Published,

Books or Other One-time Publications

Web/Internet Site

URL(s): http://people.math.gatech.edu/~dmargalit7/ Description: This site contains a list of my publications, with links to pdf files. The theorems mentioned in the results section can be found here, as well as drafts of my two books. Also, the notes for my knot theory course.

Other Specific Products

Contributions Contributions within Discipline: Bestvina, Bux, and I answered several outstanding questions in the field of Geometric Group Theory. Allen Hatcher has written a paper that relates to our work. The paper with Saul Schleimer quickly spawned one followup paper by another group. The paper with Leininger completely answers a well-known question of Luis Paris. The second paper with Farb and Leininger introduces a new phenomenon in the study of low-dilatation pseudo-Anosov mapping classes, already being investigated by others.

Page 4 of 5 Final Report: 1122020 The work with Brendle, Hatcher, and Putman contributes to a question of Dick Hain. The most recent paper with Leininger gives a new perspective on small entropy pseudo-Anosov homeomorphisms, to complement the algebraic and 3-manifold perspectives discovered with Benson Farb.

Contributions to Other Disciplines:

Contributions to Human Resource Development:

Contributions to Resources for Research and Education:

Contributions Beyond Science and Engineering:

Conference Proceedings

Categories for which nothing is reported: Organizational Partners Any Book Any Product Contributions: To Any Other Disciplines Contributions: To Any Human Resource Development Contributions: To Any Resources for Research and Education Contributions: To Any Beyond Science and Engineering Any Conference

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