Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Edge sliding and ergodic hyperfinite decomposition   We use edge slidings and saturated disjoint Borel families to give a
conceptually simple proof of Hjorth's theorem on cost attained: if a countable
p.m.p. ergodic equivalence relation $E$ is treeable and has cost $n \in
\mathbb{N} \cup \{\infty\}$ then it is induced by an a.e. free p.m.p. action of
the free group $\mathbb{F}_n$ on $n$ generators. More importantly, our
techniques give a significant strengthening of this theorem: the action of
$\mathbb{F}_n$ can be arranged so that each of the $n$ generators alone acts
ergodically.
  The existence of an ergodic action for the first generator immediately
follows from a powerful theorem of Tucker-Drob, whose proof however uses a
recent substantial result in probability theory as a black box. We give a
constructive and purely descriptive set theoretic proof of a weaker version of
Tucker-Drob's theorem, which is enough for many of its applications, including
our strengthening of Hjorth's theorem. Our proof uses new tools, such as
asymptotic means on graphs, packed disjoint Borel families, and a cost
threshold for finitizing the connected components of nonhyperfinite graphs.

33
The categories of this paper are: plasm-ph
35
The categories of this paper are: mtrl-th
24
The categories of this paper are: dg-ga
5
The categories of this paper are: gr-qc
30
The categories of this paper are: q-alg
25
The categories of this paper are: patt-sol
1
The categories of this paper are: cs
29
The categories of this paper are: funct-an
10
The categories of this paper are: q-bio
31
The categories of this paper are: ao-sci
15
The categories of this paper are: nucl-ex
28
The categories of this paper are: alg-geom
6
The categories of this paper are: astro-ph
37
The categories of this paper are: supr-con
0
The categories of this paper are: hep-ph
12
The categories of this paper are: nucl-th