1,546 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
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
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