Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A combination theorem for affine tree-free groups   Let $\Lambda_0$ be an ordered abelian group. We show how an
$\mathrm{ATF}(\mathbb{Z}\times\Lambda_0)$ group -- that is, a group admitting a
free affine action without inversions on a $\mathbb{Z}\times\Lambda_0$-tree --
admits a natural graph of groups decomposition, where vertex groups inherit
actions on $\Lambda_0$-trees. Using recent work of various authors, it follows
that a finitely generated group admitting a free affine action on a
$\mathbb{Z}^n$-tree where no line has its orientation reversed is relatively
hyperbolic with nilpotent parabolics, is locally quasiconvex, and has solvable
word, conjugacy and isomorphism problems.
  Conversely, given a graph of groups satisfying certain conditions, we show
how an affine action of its fundamental group can be constructed. Specialising
to the case of free affine actions, we obtain a large class of
$\mathrm{ATF}(\mathbb{Z}\times\Lambda_0)$ groups that do not act freely by
isometries on any $\Lambda_1$-tree. We also give an example of a group that
admits a free isometric action on a $\mathbb{Z}\times\mathbb{Z}$-tree but which
is not residually nilpotent.

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The categories of this paper are: mtrl-th
16
The categories of this paper are: stat
1
The categories of this paper are: cs
17
The categories of this paper are: q-fin
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The categories of this paper are: atom-ph
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The categories of this paper are: plasm-ph
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The categories of this paper are: chem-ph
4
The categories of this paper are: cond-mat
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The categories of this paper are: quant-ph
30
The categories of this paper are: q-alg
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The categories of this paper are: physics
0
The categories of this paper are: hep-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: astro-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: ao-sci