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I'm a last year undergraduate student and I have taken a graduate course in geometric group theory.

I'd like to start reading some more advanced stuff in geometric group theory and in particular about automorphisms of free groups. (In specific I'm interested in $\operatorname{Out}(F_n)$.)

Could you please suggest me material (books, notes, papers)? I've started studying "The topology of $\mathrm{Out}(F_n)$" by Mladen Bestvina. Are there any similar works (or more introductory ones)?

If this is not the place for such a question please let me know.

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    $\begingroup$ I can recommend the book "Topics in Geometric Group Theory" by de la Harpe. It has tons of examples, worked out carefully. This is about geometric group theory in general, not about Out(F_n). $\endgroup$
    – nikola karabatic
    Commented Sep 26, 2019 at 12:17
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    $\begingroup$ @nikola karabatic Thanks! I knew that. I'm looking for material related to $Out(F_n)$ $\endgroup$
    – 1123581321
    Commented Sep 26, 2019 at 12:27
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    $\begingroup$ $Out(F_n)$ is the group of homotopy classes of self-homotopy equivalences of a graph with one vertex and $n$ edges. It might be helpful to know enough algebraic topology to understand this statement. Chapters 0 and 1 of Hatcher's book would be a good place to start. $\endgroup$
    – Mark Grant
    Commented Sep 26, 2019 at 12:42
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    $\begingroup$ You may want to check out Office Hours with a Geometric Group Theorist. The book goes through a variety of different topics and ideas within the field at an undergraduate level. $\endgroup$
    – Santana Afton
    Commented Sep 27, 2019 at 3:02
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    $\begingroup$ @Santana Afton Thanks! In fact that is what I'm doing the last 5 days, but it has very few things about $Out(F_n)$. I'm looking for something more advanced. $\endgroup$
    – 1123581321
    Commented Sep 27, 2019 at 7:28

3 Answers 3

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Here are some assorted recommendations.

  • Stallings's "Topology of Finite Graphs" and Bestvina's course notes "Folding Graphs and Applications" are a great introduction to a technique that has found wide-ranging applications, both in the study of $\operatorname{Out}(F_n)$ and more broadly.
  • Vogtmann's "What Is Outer Space" makes a great companion to the Bestvina survey.
  • Culler and Vogtmann's "Moduli of Graphs and Automorphisms of Free Groups" is maybe the beginning of the modern perspective on $\operatorname{Out}(F_n)$ and also a really beautiful paper.
  • Bestvina and Handel's "Train Tracks and Automorphisms of Free Groups," with the Culler–Vogtmann paper, are perhaps the biggest source of tools for the study of $\operatorname{Out}(F_n)$. The later series on "The Tits Alternative for $\operatorname{Out}(F_n)$" with Feighn and the more recent Feighn–Handel "The Recognition Theorem for $\operatorname{Out}(F_n)$" build on this technology.

The above are sort of what I would consider the backbone of the theory. From there are the various hyperbolic complexes associated to $\operatorname{Out}(F_n)$, there's the study of geodesic currents or the bordification of Outer Space, and numerous other papers and related topics that grew out of this study.

$\operatorname{Out}(F_n)$ is a very interesting group in part because many people who study it do so by ruthlessly analogizing with the mapping class group of a surface or with linear groups in order to get a feeling for what ought or ought not to be the case. I think in order to understand the perspective of people working in the field, it helps to be open to broadening your understanding of other areas.

Actually, that last sentence applies to geometric group theory writ large: the field grew out of tools and insights developed to capture the essence of behaviour that people noticed in a wide variety of subfields of group theory, geometry and topology. Yours and my generation is peculiar in part because we began by studying geometric group theory as its own discipline. I'm not sure anything is lost here per se, but it can help to understand the perspective of the people whose work ours builds off of.

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  • $\begingroup$ Thank you very much! When you say 'Bestvina's course notes "Folding Graphs and Applications" ' you mean a pdf with the titlie "Folding graphs and applications, d’apr`es Stallings"? $\endgroup$
    – 1123581321
    Commented Sep 29, 2019 at 17:45
  • $\begingroup$ @giannispapav yes! that's the one. I found the process of thinking through all the exercises very helpful. $\endgroup$
    – Robbie Lyman
    Commented Sep 29, 2019 at 17:46
  • $\begingroup$ Thanks again !! $\endgroup$
    – 1123581321
    Commented Sep 29, 2019 at 17:48
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I have course notes available on Out(F_n): https://websites.umich.edu/~alexmw/Math636Notes.pdf

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I'm obliged to add "The topology, geometry, and dynamics of free groups. Part I: Outer space, fold paths, and the Nielsen/Whitehead problems" by Lee Mosher which I find very helpful (looking forward to Parts II & III).

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