Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Homogeneity and prime models in torsion-free hyperbolic groups   We show that any nonabelian free group $F$ of finite rank is homogeneous;
that is for any tuples $\bar a$, $\bar b \in F^n$, having the same complete
$n$-type, there exists an automorphism of $F$ which sends $\bar a$ to $\bar b$.
  We further study existential types and we show that for any tuples $\bar a,
\bar b \in F^n$, if $\bar a$ and $\bar b$ have the same existential $n$-type,
then either $\bar a$ has the same existential type as a power of a primitive
element, or there exists an existentially closed subgroup $E(\bar a)$ (resp.
$E(\bar b)$) of $F$ containing $\bar a$ (resp. $\bar b$) and an isomorphism
$\sigma : E(\bar a) \to E(\bar b)$ with $\sigma(\bar a)=\bar b$.
  We will deal with non-free two-generated torsion-free hyperbolic groups and
we show that they are $\exists$-homogeneous and prime. This gives, in
particular, concrete examples of finitely generated groups which are prime and
not QFA.

31
The categories of this paper are: ao-sci
12
The categories of this paper are: nucl-th
17
The categories of this paper are: q-fin
22
The categories of this paper are: chao-dyn
11
The categories of this paper are: hep-lat
20
The categories of this paper are: acc-phys
30
The categories of this paper are: q-alg
14
The categories of this paper are: math-ph
13
The categories of this paper are: quant-ph
35
The categories of this paper are: mtrl-th
0
The categories of this paper are: hep-ph
16
The categories of this paper are: stat
7
The categories of this paper are: hep-th
27
The categories of this paper are: comp-gas
29
The categories of this paper are: funct-an
26
The categories of this paper are: bayes-an