Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Large Scale Geometry of Nilpotent-by-Cyclic Groups   A nonpolycyclic nilpotent-by-cyclic group Gamma can be expressed as the HNN
extension of a finitely-generated nilpotent group N. The first main result is
that quasi-isometric nilpotent-by-cyclic groups are HNN extensions of
quasi-isometric nilpotent groups. The nonsurjective injection defining such an
extension induces an injective endomorphism phi of the Lie algebra g associated
to the Lie group in which N is a lattice. A normal form for automorphisms of
nilpotent Lie algebras--permuted absolute Jordan form-- is defined and
conjectured to be a quasi-isometry invariant. We show that if phi, theta are
endomorphisms of lattices in a fixed Carnot group G, and if the induced
automorphisms of g have the same permuted absolute Jordan form, then Gamma_phi
and Gamma_theta are quasi-isometric. Two quasi-isometry invariants are also
found: the set of ``divergence rates'' of vertical flow lines: D_phi the
``growth spaces'': g_n subset g These do not establish that permuted absolute
Jordan form is a quasi-isometry invariant, although they are major steps toward
that conjecture. Furthermore, the quasi-isometric rigidity of
finitely-presented nilpotent-by-cyclic groups is proven: any finitely-presented
group quasi-isometric to a nonpolycyclic nilpotent-by-cyclic group is
(virtually-nilpotent)-by-cyclic.

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The categories of this paper are: chem-ph
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The categories of this paper are: eess
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The categories of this paper are: cs
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The categories of this paper are: math-ph
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The categories of this paper are: physics
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The categories of this paper are: solv-int
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The categories of this paper are: plasm-ph
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The categories of this paper are: q-alg
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The categories of this paper are: hep-lat
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The categories of this paper are: cmp-lg
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The categories of this paper are: mtrl-th
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The categories of this paper are: alg-geom
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The categories of this paper are: q-bio
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The categories of this paper are: ao-sci
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The categories of this paper are: hep-ex
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The categories of this paper are: econ