Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Random triangular groups at density 1/3   Let \Gamma(n,p) denote the binomial model of a random triangular group. We
show that there exist constants c, C > 0 such that if p <= c/n^2, then a.a.s.
\Gamma(n,p) is free and if p >= C log n/n^2 then a.a.s. \Gamma(n,p) has
Kazhdan's property (T). Furthermore, we show that there exist constants C',c' >
0 such that if C'/n^2 <= p <= c' log n/n^2, then a.a.s. \Gamma(n,p) is neither
free nor has Kazhdan's property (T).

15
The categories of this paper are: nucl-ex
7
The categories of this paper are: hep-th
37
The categories of this paper are: supr-con
3
The categories of this paper are: physics
12
The categories of this paper are: nucl-th
17
The categories of this paper are: q-fin
6
The categories of this paper are: astro-ph
22
The categories of this paper are: chao-dyn
25
The categories of this paper are: patt-sol
4
The categories of this paper are: cond-mat
36
The categories of this paper are: cmp-lg
27
The categories of this paper are: comp-gas
11
The categories of this paper are: hep-lat
23
The categories of this paper are: solv-int
8
The categories of this paper are: hep-ex
19
The categories of this paper are: eess