431 research outputs found
Conjugacy in Houghton's Groups
Let . Houghton's group is the group of permutations of
, that eventually act as a translation in each
copy of . We prove the solvability of the conjugacy problem and
conjugator search problem for , .Comment: 11 pages, 1 figure, v2 correct typos and fills a small gap in the
argumen
Commutators in groups of piecewise projective homeomorphisms
In 2012 Monod introduced examples of groups of piecewise projective
homeomorphisms which are not amenable and which do not contain free subgroups,
and later Lodha and Moore introduced examples of finitely presented groups with
the same property. In this article we examine the normal subgroup structure of
these groups. Two important cases of our results are the groups and .
We show that the group of piecewise projective homeomorphisms of
has the property that is simple and that every proper
quotient of is metabelian. We establish simplicity of the commutator
subgroup of the group , which admits a presentation with generators
and relations. Further we show that every proper quotient of is
abelian. It follows that the normal subgroups of these groups are in bijective
correspondence with those of the abelian (or metabelian) quotient
The conjugacy problem in extensions of Thompson's group F
The final publication is available at Springer via http://dx.doi.org/10.1007/s11856-016-1403-9We solve the twisted conjugacy problem on Thompson’s group F. We also exhibit orbit undecidable subgroups of Aut(F), and give a proof that Aut(F) and Aut+(F) are orbit decidable provided a certain conjecture on Thompson’s group T is true. By using general criteria introduced by Bogopolski, Martino and Ventura in [5], we construct a family of free extensions of F where the conjugacy problem is unsolvable. As a byproduct of our techniques, we give a new proof of a result of Bleak–Fel’shtyn–Gonçalves in [4] showing that F has property R8, and which can be extended to show that Thompson’s group T also has property R8.Peer ReviewedPostprint (author's final draft
Commensurations and Metric Properties of Houghton's Groups
We describe the automorphism groups and the abstract commensurators of
Houghton's groups. Then we give sharp estimates for the word metric of these
groups and deduce that the commensurators embed into the corresponding
quasi-isometry groups. As a further consequence, we obtain that the Houghton
group on two rays is at least quadratically distorted in those with three or
more rays
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