Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Spheres and Projections for $\mathrm{Out}(F_n)$   The outer automorphism group Out(F_2g) of a free group on 2g generators
naturally contains the mapping class group of a punctured surface as a
subgroup. We define a subsurface projection of the sphere complex of the
connected sum of n copies of S^1 x S^2 into the arc complex of the surface and
use this to show that this subgroup is a Lipschitz retract of Out(F_2g). We
also use subsurface projections to give a simple proof of a result of Handel
and Mosher [HM10] stating that stabilizers of conjugacy classes of free
splittings and corank 1 free factors in a free group Fn are Lipschitz retracts
of Out(F_n).

30
The categories of this paper are: q-alg
16
The categories of this paper are: stat
6
The categories of this paper are: astro-ph
28
The categories of this paper are: alg-geom
36
The categories of this paper are: cmp-lg
5
The categories of this paper are: gr-qc
33
The categories of this paper are: plasm-ph
24
The categories of this paper are: dg-ga
9
The categories of this paper are: nlin
14
The categories of this paper are: math-ph
1
The categories of this paper are: cs
29
The categories of this paper are: funct-an
7
The categories of this paper are: hep-th
37
The categories of this paper are: supr-con
20
The categories of this paper are: acc-phys
15
The categories of this paper are: nucl-ex