527 research outputs found
Packing subgroups in solvable groups
We show that any subgroup of a (virtually) nilpotent-by-polycyclic group
satisfies the bounded packing property of Hruska-Wise. In particular, the same
is true about metabelian groups and linear solvable groups. However, we find an
example of a finitely generated solvable group of derived length 3 which admits
a finitely generated subgroup without the bounded packing property. In this
example the subgroup is a metabelian retract also. Thus we obtain a negative
answer to Problem 2.27 of Hruska-Wise. On the other hand, we show that
polycyclic subgroups of solvable groups satisfy the bounded packing property.Comment: 8 pages, no figur
Which groups are amenable to proving exponent two for matrix multiplication?
The Cohn-Umans group-theoretic approach to matrix multiplication suggests
embedding matrix multiplication into group algebra multiplication, and bounding
in terms of the representation theory of the host group. This
framework is general enough to capture the best known upper bounds on
and is conjectured to be powerful enough to prove , although
finding a suitable group and constructing such an embedding has remained
elusive. Recently it was shown, by a generalization of the proof of the Cap Set
Conjecture, that abelian groups of bounded exponent cannot prove
in this framework, which ruled out a family of potential constructions in the
literature.
In this paper we study nonabelian groups as potential hosts for an embedding.
We prove two main results:
(1) We show that a large class of nonabelian groups---nilpotent groups of
bounded exponent satisfying a mild additional condition---cannot prove in this framework. We do this by showing that the shrinkage rate of powers
of the augmentation ideal is similar to the shrinkage rate of the number of
functions over that are degree polynomials;
our proof technique can be seen as a generalization of the polynomial method
used to resolve the Cap Set Conjecture.
(2) We show that symmetric groups cannot prove nontrivial bounds on
when the embedding is via three Young subgroups---subgroups of the
form ---which is a
natural strategy that includes all known constructions in .
By developing techniques for negative results in this paper, we hope to
catalyze a fruitful interplay between the search for constructions proving
bounds on and methods for ruling them out.Comment: 23 pages, 1 figur
Approximate groups, III: the unitary case
By adapting the classical proof of Jordan's theorem on finite subgroups of
linear groups, we show that every approximate subgroup of the unitary group
U_n(C) is almost abelian.Comment: 19 pages, several revisions in the light of very helpful comments
from the referee, including a simplification of the main argumen
Approximate groups and their applications: work of Bourgain, Gamburd, Helfgott and Sarnak
This is a survey of several exciting recent results in which techniques
originating in the area known as additive combinatorics have been applied to
give results in other areas, such as group theory, number theory and
theoretical computer science. We begin with a discussion of the notion of an
approximate group and also that of an approximate field, describing key results
of Freiman-Ruzsa, Bourgain-Katz-Tao, Helfgott and others in which the structure
of such objects is elucidated. We then move on to the applications. In
particular we will look at the work of Bourgain and Gamburd on expansion
properties of Cayley graphs on SL_2(F_p) and at its application in the work of
Bourgain, Gamburd and Sarnak on nonlinear sieving problems.Comment: 25 pages. Survey article to accompany my forthcoming talk at the
Current Events Bulletin of the AMS, 2010. A reference added and a few small
changes mad
On the asymptotic geometry of abelian-by-cyclic groups
A finitely presented, torsion free, abelian-by-cyclic group can always be
written as an ascending HNN extension Gamma_M of Z^n, determined by an n x n
integer matrix M with det(M) \ne 0. The group Gamma_M is polycyclic if and only
if |det(M)|=1. We give a complete classification of the nonpolycyclic groups
Gamma_M up to quasi-isometry: given n x n integer matrices M,N with |det(M)|,
|det(N)| > 1, the groups Gamma_M, Gamma_N are quasi-isometric if and only if
there exist positive integers r,s such that M^r, N^s have the same absolute
Jordan form. We also prove quasi-isometric rigidity: if Gamma_M is an
abelian-by-cyclic group determined by an n x n integer matrix M with |det(M)| >
1, and if G is any finitely generated group quasi-isometric to Gamma_M, then
there is a finite normal subgroup K of G such that G/K is abstractly
commensurable to Gamma_N, for some n x n integer matrix N with |det(N)| > 1.Comment: 65 pages, 2 figures. To appear in Acta Mathematic
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
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