Connections of Geometric Group Theory with other areas of mathematics.

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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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).

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. Note also that my mathoverflow answer gives a reference to the use of groupoids in PDEs.

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.