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Abstract
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The main goal of this note is to suggest an algebraic approach to the quasi-isometric
classification of partially commutative groups (alias right-angled Artin groups).
More precisely, we conjecture that if the partially commutative groups
and
are quasi-isometric,
then
is a (nice)
subgroup of
and vice-versa. We show that the conjecture holds for all known cases of
quasi-isometric classification of partially commutative groups, namely for the classes of
–trees
and atomic graphs. As in the classical Mostow rigidity theory for irreducible lattices,
we relate the quasi-isometric rigidity of the class of atomic partially commutative
groups with the algebraic rigidity, that is, with the co-Hopfian property of their
–completions.
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Keywords
partially commutative group, right-angled Artin group,
embeddability, quasi-isometric classification
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Mathematical Subject Classification 2010
Primary: 20A15, 20F36, 20F65, 20F69
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Publication
Received: 4 March 2015
Revised: 9 June 2015
Accepted: 5 July 2015
Published: 23 February 2016
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