2,914 research outputs found
Some characterizations of Howson PC-groups
We show that in the class of partially commutative groups, the conditions of
being Howson, being fully residually free, and being free product of
free-abelian groups, are equivalent
On the lattice of subgroups of a free group: complements and rank
A -complement of a subgroup is a subgroup such that . If we also ask
to have trivial intersection with , then we say that is a
-complement of . The minimum possible rank of a -complement
(resp. -complement) of is called the -corank (resp.
-corank) of . We use Stallings automata to study these notions and
the relations between them. In particular, we characterize when complements
exist, compute the -corank, and provide language-theoretical descriptions
of the sets of cyclic complements. Finally, we prove that the two notions of
corank coincide on subgroups that admit cyclic complements of both kinds.Comment: 27 pages, 5 figure
Intersection problem for Droms RAAGs
We solve the subgroup intersection problem (SIP) for any RAAG G of Droms type
(i.e., with defining graph not containing induced squares or paths of length
3): there is an algorithm which, given finite sets of generators for two
subgroups H,K of G, decides whether is finitely generated or not,
and, in the affirmative case, it computes a set of generators for .
Taking advantage of the recursive characterization of Droms groups, the proof
consists in separately showing that the solvability of SIP passes through free
products, and through direct products with free-abelian groups. We note that
most of RAAGs are not Howson, and many (e.g. F_2 x F_2) even have unsolvable
SIP.Comment: 33 pages, 12 figures (revised following the referee's suggestions
Small-world behavior in a system of mobile elements
We analyze the propagation of activity in a system of mobile automata. A
number r L^d of elements move as random walkers on a lattice of dimension d,
while with a small probability p they can jump to any empty site in the system.
We show that this system behaves as a Dynamic Small-World (DSW) and present
analytic and numerical results for several quantities. Our analysis shows that
the persistence time T* (equivalent to the persistence size L* of small-world
networks) scales as T* ~ (r p)^(-t), with t = 1/(d+1).Comment: To appear in Europhysics Letter
Algorithmic recognition of infinite cyclic extensions
We prove that one cannot algorithmically decide whether a finitely presented
-extension admits a finitely generated base group, and we use this
fact to prove the undecidability of the BNS invariant. Furthermore, we show the
equivalence between the isomorphism problem within the subclass of unique
-extensions, and the semi-conjugacy problem for deranged outer
automorphisms.Comment: 24 page
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