45,156 research outputs found
Brauer relations in finite groups
If G is a non-cyclic finite group, non-isomorphic G-sets X, Y may give rise
to isomorphic permutation representations C[X] and C[Y]. Equivalently, the map
from the Burnside ring to the representation ring of G has a kernel. Its
elements are called Brauer relations, and the purpose of this paper is to
classify them in all finite groups, extending the Tornehave-Bouc classification
in the case of p-groups.Comment: 39 pages; final versio
Non-elementary amenable subgroups of automata groups
We consider groups of automorphisms of locally finite trees, and give
conditions on its subgroups that imply that they are not elementary amenable.
This covers all known examples of groups that are not elementary amenable and
act on the trees: groups of intermediate growths and Basilica group, by giving
a more straightforward proof. Moreover, we deduce that all finitely generated
branch groups are not elementary amenable, which was conjectured by Grigorchuk
Totally disconnected locally compact groups locally of finite rank
We study totally disconnected locally compact second countable (t.d.l.c.s.c.)
groups that contain a compact open subgroup with finite rank. We show such
groups that additionally admit a pro- compact open subgroup for some
finite set of primes are virtually an extension of a finite direct
product of topologically simple groups by an elementary group. This result, in
particular, applies to l.c.s.c. -adic Lie groups. We go on to prove a
decomposition result for all t.d.l.c.s.c. groups containing a compact open
subgroup with finite rank. In the course of proving these theorems, we
demonstrate independently interesting structure results for t.d.l.c.s.c. groups
with a compact open pro-nilpotent subgroup and for topologically simple
l.c.s.c. -adic Lie groups.Comment: Referee's suggestions incorporated. Main theorems for the general
locally pro-nilpotent and the general locally of finite rank cases improve
Deciding Isomorphy using Dehn fillings, the splitting case
We solve Dehn's isomorphism problem for virtually torsion-free relatively
hyperbolic groups with nilpotent parabolic subgroups.
We do so by reducing the isomorphism problem to three algorithmic problems in
the parabolic subgroups, namely the isomorphism problem, separation of torsion
(in their outer automorphism groups) by congruences, and the mixed Whitehead
problem, an automorphism group orbit problem. The first step of the reduction
is to compute canonical JSJ decompositions. Dehn fillings and the given
solutions of the algorithmic problems in the parabolic groups are then used to
decide if the graphs of groups have isomorphic vertex groups and, if so,
whether a global isomorphism can be assembled.
For the class of finitely generated nilpotent groups, we give solutions to
these algorithmic problems by using the arithmetic nature of these groups and
of their automorphism groups.Comment: 76 pages. This version incorporates referee comments and corrections.
The main changes to the previous version are a better treatment of the
algorithmic recognition and presentation of virtually cyclic subgroups and a
new proof of a rigidity criterion obtained by passing to a torsion-free
finite index subgroup. The previous proof relied on an incorrect result. To
appear in Inventiones Mathematica
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.
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