Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Properness of nilprogressions and the persistence of polynomial growth
  of given degree   We show that an arbitrary nilprogression can be approximated by a proper
coset nilprogression in upper-triangular form. This can be thought of as a
nilpotent version of the Freiman-Bilu result that a generalised arithmetic
progression can be efficiently contained in a proper generalised arithmetic
progression, and indeed an important ingredient in the proof is a Lie-algebra
version of the geometry-of-numbers argument at the centre of that result. We
also present some applications. We verify a conjecture of Benjamini that if $S$
is a symmetric generating set for a group such that $1\in S$ and $|S^n|\le
Mn^D$ at some sufficiently large scale $n$ then $S$ exhibits polynomial growth
of the same degree $D$ at all subsequent scales, in the sense that
$|S^r|\ll_{M,D}r^D$ for every $r\ge n$. Our methods also provide an important
ingredient in a forthcoming companion paper in which we reprove and sharpen a
result about scaling limits of vertex-transitive graphs of polynomial growth
due to Benjamini, Finucane and the first author. We also note that our
arguments imply that every approximate group has a large subset with a large
quotient that is Freiman isomorphic to a subset of a torsion-free nilpotent
group of bounded rank and step.

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The categories of this paper are: stat
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The categories of this paper are: ao-sci
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The categories of this paper are: mtrl-th
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The categories of this paper are: cond-mat
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The categories of this paper are: atom-ph
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The categories of this paper are: nlin
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The categories of this paper are: cmp-lg
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The categories of this paper are: chao-dyn
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The categories of this paper are: quant-ph
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The categories of this paper are: adap-org
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The categories of this paper are: plasm-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: acc-phys
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The categories of this paper are: supr-con
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The categories of this paper are: math-ph
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The categories of this paper are: gr-qc