Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the spectral theory of groups of automorphisms of $S$-adic
  nilmanifolds   Let $S=\{p_1, \dots, p_r,\infty\}$ for prime integers $p_1, \dots, p_r.$ Let
$X$ be an $S$-adic compact nilmanifold, equipped with the unique translation
invariant probability measure $\mu.$ We characterize the countable groups
$\Gamma$ of automorphisms of $X$ for which the Koopman representation $\kappa$
on $L^2(X,\mu)$ has a spectral gap. More specifically, we show that $\kappa$
does not have a spectral gap if and only if there exists a non-trivial
$\Gamma$-invariant quotient solenoid (that is, a finite-dimensional, connected,
compact abelian group) on which $\Gamma$ acts as a virtually abelian group.

17
The categories of this paper are: q-fin
18
The categories of this paper are: econ
29
The categories of this paper are: funct-an
19
The categories of this paper are: eess
15
The categories of this paper are: nucl-ex
14
The categories of this paper are: math-ph
5
The categories of this paper are: gr-qc
30
The categories of this paper are: q-alg
37
The categories of this paper are: supr-con
3
The categories of this paper are: physics
33
The categories of this paper are: plasm-ph
7
The categories of this paper are: hep-th
32
The categories of this paper are: atom-ph
12
The categories of this paper are: nucl-th
10
The categories of this paper are: q-bio
36
The categories of this paper are: cmp-lg