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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.

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Wouldn't we all! – HJRW Apr 17 '14 at 13:27
up vote 3 down vote accepted

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

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Yes, I have fixed it. – Dietrich Burde Apr 17 '14 at 18:48

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":

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I don't think Church/Farb found a new proof of the prime number theorem, but rather a way to count irreducible polynomials (of given degree) over $F_q,$ which was first done by Dedekind [I believe], and is a matter of simple combinatorics (it is an exercise in Knuth volume 2). Also, I would not call this Church/Farb work "geometric". – Igor Rivin Apr 17 '14 at 14:35
Ok, I modified my answer a bit. – Khalid Bou-Rabee Apr 17 '14 at 14:38

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).

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Can you give some reference/explanation for this? – Lennart Meier Apr 17 '14 at 14:48
Ji, Lizhen: A tale of two groups: arithmetic groups and mapping class groups. Handbook of Teichmüller theory. Volume III, 157–295. MR2952766 – Francesco Polizzi Apr 17 '14 at 16:43

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