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I'm a master's student in the Turin University. At the end of my studies, I have to write a master thesis. My main interest is geometric group theory, but it is not a research area of the Turin's mathematical department.

My professors in algebra and geometry are principally interested in algebraic geometry, commutative algebra or in Lie groups. My analysis professors are interested mainly in PDE and differential forms. So neither the theory of non-positive curved spaces, nor measure group theory seems to be a feasible alternative.

Since I want to do a PhD – not in Turin – I don't want to do a thesis too distant from my main interest. What I want to know is then if you know some topics of geometric group theory (or other similar theories) that are sufficiently connected with algebraic geometry (for example, with riemannian surfaces) or with Lie Groups and can be explored by a master student.

I haven't had yet time to explore deeply the main book on the subject – de la Harpe, Geoghegan, Bridson-Haefliger, Bowditch, Serre, Farb(on mapping class groups). Anyway, I suppose the theory presented in the Farb's book is the one that has more connections with the topics I listed before, or I'm wrong?

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2 Answers 2

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In view of your declared main interest, one possibility is to put the question the other way round: are there areas of mathematics which are relevant to but maybe not much exploited in geometric group theory?

My own feeling is that the work on crossed modules given an exposition in Part I of our new book

R. Brown, P.J. Higgins, R. Sivera, Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy groupoids, EMS Tracts in Mathematics Vol. 15, 703 pages. (August 2011).

(pdf available from http://pages.bangor.ac.uk/~mas010/nonab-a-t.html), is really a form of geometric group theory, but is not taken into account in the works you cite.

Crossed modules and the associated 2-groups have quite a wide literature.

This is the answer which occurs to me, but your evaluation of it is entirely up to you!

Another relevant reference is

Higgins, P.J. Notes on categories and groupoids, Mathematical Studies, Volume 32. Van Nostrand Reinhold Co. London (1971); Reprints in Theory and Applications of Categories, No. 7 (2005) pp 1--195. (downloadable)

His groupoid techniques, though dating from the 1960s, are not, it seems to me, embedded in the work on geometric group theory, though they have been used by some workers.

Hope that helps.

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Actually, I recently read, on May's introductory book, the fundamental groupoid version of the Van Kampen Theorem, and I found it very insightful. So, do you suggest to explore the theory of Crossed modules and associated 2-groups? Or do you suggest to try to explore one of the topic presented in the future directions chapter (for example 16.1.1 or try to reformulate some of the theory presented there using crossed modules and complexes)? –  Vittorio Patriarca Jun 7 '12 at 18:24
    
I would not like to be prescriptive, but suggest you look at the sections of the new book on intuitions, and on van Kampen diagrams, and see if there are possible links with the extensive work of for example Olʹshanskii, A. Yu. using diagrams. But I always told my students their job was to evaluate my suggestions, and think about why I had made them, e.g. basic intuitions, how one would like the theory to go, have A's tools been fully used in B's theory, etc.!!! At a master's level, or any level indeed, you always have to consider feasability. –  Ronnie Brown Jun 14 '12 at 20:18

Since your "analysis professors are interested mainly in PDE and differential forms", you could explore PDE topics that are relevant to geometric group theory. Gromov's theorem on groups of polynomial growth is a result of central importance in the subject, and its proof (I mean the new one, by Kleiner) is built on a PDE fact: the space of solutions of an elliptic PDE with controlled growth is finite-dimensional.

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Wonderful, a very good example of how different mathematical fields can help each other. Unfortunately, I'm not a big fan of analysis, but you surely give me a reason to study it more seriously. –  Vittorio Patriarca Jun 7 '12 at 17:17

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