Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Spectral Rigidity and Subgroups of Free Groups   A subset $\Sigma \subset F_N$ of the free group of rank $N$ is called
\emph{spectrally rigid} if whenever trees $T, T'$ in Culler-Vogtmann Outer
Space are such that $\| g \|_T = \| g \|_{T'}$ for every $g \in \Sigma$, it
follows that $T = T'$. Results of Smillie, Vogtmann, Cohen, Lustig, and Steiner
prove that (for $N \geq 2$) no finite subset of $F_N$ is spectrally rigid in
$F_N$. We prove that if $\{ H_i \}_{i=1}^k$ is a finite collection of
subgroups, each of infinite index, and $g_i \in F_N$, then $\cup_{i=1}^k g_i
H_i$ is not spectrally rigid in $F_N$. Taking $H_i = 1$, we recover the results
about finite sets. We also prove that any coset of a nontrivial normal subgroup
$H \lhd F_N$ is spectrally rigid.

9
The categories of this paper are: nlin
8
The categories of this paper are: hep-ex
32
The categories of this paper are: atom-ph
33
The categories of this paper are: plasm-ph
27
The categories of this paper are: comp-gas
5
The categories of this paper are: gr-qc
1
The categories of this paper are: cs
36
The categories of this paper are: cmp-lg
14
The categories of this paper are: math-ph
12
The categories of this paper are: nucl-th
22
The categories of this paper are: chao-dyn
16
The categories of this paper are: stat
15
The categories of this paper are: nucl-ex
20
The categories of this paper are: acc-phys
13
The categories of this paper are: quant-ph
25
The categories of this paper are: patt-sol