223 research outputs found
Inverse monoids and immersions of 2-complexes
It is well known that under mild conditions on a connected topological space
, connected covers of may be classified via conjugacy
classes of subgroups of the fundamental group of . In this paper,
we extend these results to the study of immersions into 2-dimensional
CW-complexes. An immersion between
CW-complexes is a cellular map such that each point has a
neighborhood that is mapped homeomorphically onto by . In order
to classify immersions into a 2-dimensional CW-complex , we need to
replace the fundamental group of by an appropriate inverse monoid.
We show how conjugacy classes of the closed inverse submonoids of this inverse
monoid may be used to classify connected immersions into the complex
On subgroups of R. Thompson's group
We provide two ways to show that the R. Thompson group has maximal
subgroups of infinite index which do not fix any number in the unit interval
under the natural action of on , thus solving a problem by D.
Savchuk. The first way employs Jones' subgroup of the R. Thompson group and
leads to an explicit finitely generated example. The second way employs
directed 2-complexes and 2-dimensional analogs of Stallings' core graphs, and
gives many implicit examples. We also show that has a decreasing sequence
of finitely generated subgroups such that and
for every there exist only finitely many subgroups of containing .Comment: 20 pages; v2: fixed some misprints, filled a gap in the proof of
Theorem 4.1, added Remark 4.1 that Homeo^+(R) and many subgrioups of that
group are quasi-residually finite; v3: Section 5 added, final version
accepted to Transactions of the AM
Stallings graphs for quasi-convex subgroups
We show that one can define and effectively compute Stallings graphs for
quasi-convex subgroups of automatic groups (\textit{e.g.} hyperbolic groups or
right-angled Artin groups). These Stallings graphs are finite labeled graphs,
which are canonically associated with the corresponding subgroups. We show that
this notion of Stallings graphs allows a unified approach to many algorithmic
problems: some which had already been solved like the generalized membership
problem or the computation of a quasi-convexity constant (Kapovich, 1996); and
others such as the computation of intersections, the conjugacy or the almost
malnormality problems.
Our results extend earlier algorithmic results for the more restricted class
of virtually free groups. We also extend our construction to relatively
quasi-convex subgroups of relatively hyperbolic groups, under certain
additional conditions.Comment: 40 pages. New and improved versio
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
- …