Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Direct products, overlapping actions, and critical regularity   We address the problem of computing the critical regularity of groups of
homeomorphisms of the interval. Our main result is that if $H$ and $K$ are two
non-solvable groups then a faithful $C^{1,\tau}$ action of $H\times K$ on a
compact interval $I$ is {\em not overlapping} for all $\tau>0$, which by
definition means that there must be non-trivial $h\in H$ and $k\in K$ with
disjoint support. As a corollary we prove that the right-angled Artin group
$(F_2\times F_2)*\mathbb{Z}$ has critical regularity one, which is to say that
it admits a faithful $C^1$ action on $I$, but no faithful $C^{1,\tau}$ action.
This is the first explicit example of a group of exponential growth which is
without nonabelian subexponential growth subgroups, whose critical regularity
is finite, achieved, and known exactly. Another corollary we get is that
Thompson's group $F$ does not admit a faithful $C^1$ overlapping action on $I$,
so that $F*\mathbb{Z}$ is a new example of a locally indicable group admitting
no faithful $C^1$--action on $I$.

11
The categories of this paper are: hep-lat
7
The categories of this paper are: hep-th
20
The categories of this paper are: acc-phys
1
The categories of this paper are: cs
36
The categories of this paper are: cmp-lg
16
The categories of this paper are: stat
35
The categories of this paper are: mtrl-th
31
The categories of this paper are: ao-sci
32
The categories of this paper are: atom-ph
27
The categories of this paper are: comp-gas
30
The categories of this paper are: q-alg
0
The categories of this paper are: hep-ph
28
The categories of this paper are: alg-geom
6
The categories of this paper are: astro-ph
25
The categories of this paper are: patt-sol
8
The categories of this paper are: hep-ex