2,367 research outputs found
A proof that all Seifert 3-manifold groups and all virtual surface groups are conjugacy separable
We prove that the fundamental group of any Seifert 3-manifold is conjugacy
separable. That is, conjugates may be distinguished in finite quotients or,
equivalently, conjugacy classes are closed in the pro-finite topology.Comment: 8 page
Conjugacy in normal subgroups of hyperbolic groups
Let N be a finitely generated normal subgroup of a Gromov hyperbolic group G.
We establish criteria for N to have solvable conjugacy problem and be conjugacy
separable in terms of the corresponding properties of G/N. We show that the
hyperbolic group from F. Haglund's and D. Wise's version of Rips's construction
is hereditarily conjugacy separable. We then use this construction to produce
first examples of finitely generated and finitely presented conjugacy separable
groups that contain non-(conjugacy separable) subgroups of finite index.Comment: Version 3: 18 pages; corrected a problem with justification of
Corollary 8.
Stretching factors, metrics and train tracks for free products
In this paper we develop the metric theory for the outer space of a free
product of groups. This generalizes the theory of the outer space of a free
group, and includes its relative versions. The outer space of a free product is
made of -trees with possibly non-trivial vertex stabilisers. The strategies
are the same as in the classical case, with some technicalities arising from
the presence of infinite-valence vertices.
In particular, we describe the Lipschitz metric and show how to compute it;
we prove the existence of optimal maps; we describe geodesics represented by
folding paths. We show that train tracks representative of irreducible (hence
hyperbolic) automorphisms exist and that their are metrically characterized as
minimal displaced points, showing in particular that the set of train tracks is
closed. We include a proof of the existence of simplicial train tracks map
without using Perron-Frobenius theory.
A direct corollary of this general viewpoint is an easy proof that relative
train track maps exist in both the free group and free product case.Comment: Article updated with minor revision
Graph-wreath products and finiteness conditions
A notion of \emph{graph-wreath product} is introduced. We obtain sufficient
conditions for these products to satisfy the topologically inspired finiteness
condition type . Under various additional assumptions we
show that these conditions are necessary. Our results generalise results of
Cornulier about wreath products in case . Graph-wreath products include
classical permutational wreath products and semidirect products of right-angled
Artin groups by groups of automorphisms amongst others.Comment: 12 page
Orbit decidability and the conjugacy problem for some extensions of groups
Given a short exact sequence of groups with certain conditions, 1 ? F ? G ? H ? 1, weprove that G has solvable conjugacy problem if and only if the corresponding action subgroupA 6 Aut(F) is orbit decidable. From this, we deduce that the conjugacy problem is solvable,among others, for all groups of the form Z2?Fm, F2?Fm, Fn?Z, and Zn?A Fm with virtually solvable action group A 6 GLn(Z). Also, we give an easy way of constructing groups of the form Z4?Fn and F3?Fn with unsolvable conjugacy problem. On the way, we solve the twisted conjugacy problem for virtually surface and virtually polycyclic groups, and give an example of a group with solvable conjugacy problem but unsolvable twisted conjugacy problem. As an application, an alternative solution to the conjugacy problem in Aut(F2) is given
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
Degree of commutativity of infinite groups
First published in Proceedings of the American Mathematical Society in volum 145, number 2, 2016, published by the American Mathematical SocietyWe prove that, in a finitely generated residually finite group of subexponential growth, the proportion of commuting pairs is positive if and only if the group is virtually abelian. In particular, this covers the case where the group has polynomial growth (i.e., virtually nilpotent groups, where the hypothesis of residual finiteness is always satisfied). We also show that, for non-elementary hyperbolic groups, the proportion of commuting pairs is always zero.Peer ReviewedPostprint (author's final draft
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