4,488 research outputs found
Groups with context-free co-word problem
The class of co-context-free groups is studied. A co-context-free group is defined as one whose coword
problem (the complement of its word problem) is context-free. This class is larger than the
subclass of context-free groups, being closed under the taking of finite direct products, restricted
standard wreath products with context-free top groups, and passing to finitely generated subgroups
and finite index overgroups. No other examples of co-context-free groups are known. It is proved
that the only examples amongst polycyclic groups or the BaumslagâSolitar groups are virtually
abelian. This is done by proving that languages with certain purely arithmetical properties cannot
be context-free; this result may be of independent interest
The coarse classification of countable abelian groups
We prove that two countable locally finite-by-abelian groups G,H endowed with
proper left-invariant metrics are coarsely equivalent if and only if their
asymptotic dimensions coincide and the groups are either both
finitely-generated or both are infinitely generated. On the other hand, we show
that each countable group G that coarsely embeds into a countable abelian group
is locally nilpotent-by-finite. Moreover, the group G is locally
abelian-by-finite if and only if G is undistorted in the sense that G can be
written as the union of countably many finitely generated subgroups G_n such
that each G_n is undistorted in G_{n+1} (which means that the identity
inclusion from G_n to G_{n+1} is a quasi-isometric embedding with respect to
word metrics).Comment: 25 pages. Longer version with new results about FCC groups, locally
finite-by-abelian groups, locally nilpotent-by-finite groups
Finitely generated groups with polynomial index growth
We prove that a finitely generated soluble residually finite group has
polynomial index growth if and only if it is a minimax group. We also show that
if a finitely generated group with PIG is residually finite-soluble then it is
a linear group.
These results apply in particular to boundedly generated groups; they imply
that every infinite BG residually finite group has an infinite linear quotient.Comment: To appear in Crelle's Journa
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