4,094 research outputs found
Coset intersection graphs for groups
Let H, K be subgroups of G. We investigate the intersection properties of
left and right cosets of these subgroups.Comment: 4 page
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
Packing subgroups in relatively hyperbolic groups
We introduce the bounded packing property for a subgroup of a countable
discrete group G. This property gives a finite upper bound on the number of
left cosets of the subgroup that are pairwise close in G. We establish basic
properties of bounded packing, and give many examples; for instance, every
subgroup of a countable, virtually nilpotent group has bounded packing. We
explain several natural connections between bounded packing and group actions
on CAT(0) cube complexes.
Our main result establishes the bounded packing of relatively quasiconvex
subgroups of a relatively hyperbolic group, under mild hypotheses. As an
application, we prove that relatively quasiconvex subgroups have finite height
and width, properties that strongly restrict the way families of distinct
conjugates of the subgroup can intersect. We prove that an infinite,
nonparabolic relatively quasiconvex subgroup of a relatively hyperbolic group
has finite index in its commensurator. We also prove a virtual malnormality
theorem for separable, relatively quasiconvex subgroups, which is new even in
the word hyperbolic case.Comment: 45 pages, 2 figures. To appear in Geom. Topol. v2: Updated to address
concerns of the referee. Added theorem that an infinite, nonparabolic
relatively quasiconvex subgroup H of a relatively hyperbolic group has finite
index in its commensurator. Added several new geometric results to Section 7.
Theorem 8.9 on packing relative to peripheral subgroups is ne
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
Fundamental domains for congruence subgroups of SL2 in positive characteristic
In this work, we construct fundamental domains for congruence subgroups of
and . Our method uses Gekeler's description of
the fundamental domains on the Bruhat- Tits tree in terms of
cosets of subgroups. We compute the fundamental domains for a number of
congruence subgroups explicitly as graphs of groups using the computer algebra
system Magma
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