372 research outputs found
Strong approximation methods in group theory, an LMS/EPSRC Short course lecture notes
These are the lecture notes for the LMS/EPSRC short course on strong
approximation methods in linear groups organized by Dan Segal in Oxford in
September 2007.Comment: v4: Corollary 6.2 corrected, added a few small remark
Strange images of profinite groups
We investigate whether a finitely generated profinite group G could have a
finitely generated infinite image. A result of Dan Segal shows that this is
impossible if G is prosoluble. We prove that such an image does not exist if G
is semisimple or nonuniversal. We also investigate the existense of dense
normal subgroups in .Comment: The results of this preprint have been superceded by
http://arxiv.org/abs/1102.3037 which answers the questions posed her
Algebraic properties of profinite groups
Recently there has been a lot of research and progress in profinite groups.
We survey some of the new results and discuss open problems. A central theme is
decompositions of finite groups into bounded products of subsets of various
kinds which give rise to algebraic properties of topological groups.Comment: This version has some references update
Rank gradient, cost of groups and the rank versus Heegaard genus problem
We study the growth of the rank of subgroups of finite index in residually
finite groups, by relating it to the notion of cost.
As a by-product, we show that the `Rank vs. Heegaard genus' conjecture on
hyperbolic 3-manifolds is incompatible with the `Fixed Price problem' in
topological dynamics
Words with few values in finite simple groups
We construct words with small image in a given finite alternating or
unimodular group. This shows that word width in these groups is unbounded in
general.Comment: 7 page
On finitely generated profinite groups I: strong completeness and uniform bounds
We prove that in every finitely generated profinite group, every subgroup of
finite index is open; this implies that the topology on such groups is
determined by the algebraic structure. This is deduced from the main result
about finite groups: let be a `locally finite' group word and
. Then there exists such that in every -generator
finite group , every element of the verbal subgroup is equal to a
product of -values.
An analogous theorem is proved for commutators; this implies that in every
finitely generated profinite group, each term of the lower central series is
closed.
The proofs rely on some properties of the finite simple groups, to be
established in Part II.Comment: 66 page
On finitely generated profinite groups II, products in quasisimple groups
We prove two results. (1) There is an absolute constant such that for any
finite quasisimple group , given 2D arbitrary automorphisms of , every
element of is equal to a product of `twisted commutators' defined by
the given automorphisms.
(2) Given a natural number , there exist and such that:
if is a finite quasisimple group with ,
are any automorphisms of , and are any
divisors of , then there exist inner automorphisms of such
that .
These results, which rely on the Classification of finite simple groups, are
needed to complete the proofs of the main theorems of Part I.Comment: 34 page
A non-LEA Sofic Group
We describe elementary examples of finitely presented sofic groups which are
not residually amenable (and thus not initially subamenable or LEA, for short).
We ask if an amalgam of two amenable groups over a finite subgroup is
residually amenable and answer this positively for some special cases,
including countable locally finite groups, residually nilpotent groups and
others.Comment: The main theorem is strengthened so that the Sofic examples are shown
to have no co-amenable LEA subgroup
Product decompositions of quasirandom groups and a Jordan type theorem
We first note that a result of Gowers on product-free sets in groups has an
unexpected consequence: If k is the minimal degree of a representation of the
finite group G, then for every subset B of G with we have
B^3 = G.
We use this to obtain improved versions of recent deep theorems of Helfgott
and of Shalev concerning product decompositions of finite simple groups, with
much simpler proofs.
On the other hand, we prove a version of Jordan's theorem which implies that
if k>1, then G has a proper subgroup of index at most ck^2 for some absolute
constant c, hence a product-free subset of size at least . This
answers a question of Gowers.Comment: 18 pages. In this third version we added an Appendix with a short
proof of Proposition
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