Links between Geometric Group Theory and Number Theory

Question

Do You know any successful applications of the geometric group theory in the number theory? GTG is my main field of interest and I would love to use it to prove new facts in the number theory.

Answer 1

Another reference showing the connections between geometric group theory and number theory is the book of A. Lubotzky on "Discrete Groups, Expanding Graphs and Invariant Measures."

There are many special topics where a connection between number theory and geometric group theory arises. As an example see http://arxiv.org/abs/0901.1458.

Answer 2

It is hard to answer this question as both of these subjects are very broad. So I'll just quote something from the introduction of Lubotzky and Segal's Subgroup Growth:

"There have also been applications outside subgroup growth: a group-theoretic characterisation of arithmetic groups with the congruence subgroup property, estimates for the number of hyperbolic manifolds with given volume, and the results mentioned above on the enumeration and classification of finite p-groups."

It might be relevant to point out that representation theory and homological stability has some surprising connections with number theory. Here is a link to a paper by Tom Church and Benson Farb (also see the references given within) "Representation theory and homological stability": http://arxiv.org/abs/1008.1368

Answer 3

Very broad question.

An example I like is the following: the classical Minkowski reduction theory for quadratic forms can be rephrased and generalized in terms of the action of the mapping class group $\textrm{Mod}_g$ on the Teichmuller space $\mathcal{T}_g$.

Roughly speaking, one can more generally interpret reduction theories of forms in terms of fundamental domains for group actions (Siegel, Borel, Harish-Chandra and others).