Intereting Posts

A circle has the same center as an ellipse and passes through the foci $F_1$ and $F_2$ of the ellipse, two curves intersect in $4$ points.
Are there other power series for the Lambert W function than this one?
diagonalisability of matrix few properties
Why $a^n – b^n$ is divisible by $a-b$?
Sum of every $k$th binomial coefficient.
Openness condition in Seifert-van Kampen Theorem
Divergence of Reciprocal of Sequence knowing the Asymptotic Density
Proof for power functions
diagonalize a non-normal matrix , without distinct eigenvalues
A problem about periodic functions
Writing number as sum of reciprocals of factorial
What function can be differentiated twice, but not 3 times?
Random sums of iid Uniform random variables
Show that $\mathfrak c +{\aleph_0}=\mathfrak c$ using “presenters”
Frechet derivative of square root on positive elements in some $C^*$-algebra

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.

- Are the determinantal ideals prime?
- Closure of image by polynomial of irreducible algebraic variety is also irreducible algebraic variety
- Graph of morphism , reduced scheme.
- is the empty set an (irreducible) variety?
- Zero subscheme of a section: Making computations.
- Scheme: Countable union of affine lines

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?

- “The Egg:” Bizarre behavior of the roots of a family of polynomials.
- Finite etale maps to the line minus the origin
- Presheaf image of a monomorphism of sheaves is a sheaf
- Does Hom commute with stalks for locally free sheaves?
- Relationship between very ample divisors and hyperplane sections
- Tangent sheaf of a (specific) nodal curve
- Quotient varieties
- Generic Points to the Italians
- Curve in $\mathbb{A}^3$ that cannot be defined by 2 equations
- A Hunt for a Mathematical Machine That Gives Points

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.

- Quaternion – Angle computation using accelerometer and gyroscope
- The limit of $((1+x)^{1/x} – e+ ex/2)/x^2$ as $x\to 0$
- Counting subsets containing three consecutive elements (previously Summation over large values of nCr)
- Derivative of $x^{x^{\cdot^{\cdot}}}$?
- product of densities
- Are Hilbert primes also Hilbert irreducible ? Furthermore, are Hilbert primes also primes in $\mathbb{ Z}$?
- Two different formulas for standard error of difference between two means
- Why are the continued fractions of non-square-root numbers ($\sqrt{x}$ where $a>2$) not periodic?
- Circle Packing Algorithm
- Proving that an integral is differentiable
- Closed Bounded but not compact Subset of a Normed Vector Space
- Compact space and Hausdorff space
- Max distance between a line and a parabola
- Interpreting higher order differentials
- Proof of $\sum_{n=1}^{\infty}\frac1{n^3}\frac{\sinh\pi n\sqrt2-\sin\pi n\sqrt2}{{\cosh\pi n\sqrt2}-\cos\pi n\sqrt2}=\frac{\pi^3}{18\sqrt2}$