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On conjugate chains of subgroups in residually finite groups
We show that conjugate descending chains of finite index subgroups can have different intersections in a residually finite discrete group. In fact, we can obtain an arbitrary countable set of residually finite groups as the intersection set of a conjugacy class of such chains.
with L. Babai, Finite groups of uniform logarithmic diameter, to appear
in Israel J. of Math. PDF
We provide an example for a sequence of finite groups $G_n$ such that all $G_n$ can be generated by 2 elements and the diameter of any Cayley graph of $G_n$ is $Clog(G_n)$. This answers a question of Lubotzky.
On the probability of satisfying a word in a group, to appear in J.
of Group Theory PDF
We show that for any finite group
$G$ and for any $d$ there exists a word $w\in F_{d}$ such that a $d$-tuple in
$G$ satisfies $w$ if and only if it generates a solvable subgroup. In
particular, if $G$ itself is not solvable, then it cannot be obtained as a
quotient of the one relator group $F_{d}/<w>$.
As a corollary, the probability that a word is satisfied in a fixed
non-solvable group can be made arbitrarily small, answering a question of Alon
Amit.
with Y. Glasner, Generic groups acting on trees
Let A be the automorphism group of a regular tree and let G be the subgroup generated by k random elements in A. We show that almost surely G is free of rank k and every nonidentity element of G has 0 or 2 fixpoints on the boundary of the tree. Also, for k > 1 almost surely the closure of G is discrete, compact, equal to A or has index 2 in A. All four possibilities have infinite measure in $A^k$.
Group laws and free subgroups
in topological groups, Bull.
We prove that a permutation group
in which different finite sets have different stabilizers cannot satisfy any
group law. For locally compact topological groups with this property we show
that almost all finite subsets of the group generate free subgroups.
We derive consequences of these theorems on Thompson's group F, weakly branch
groups, automorphism groups of regular trees and profinite groups with
alternating composition factors of unbounded degree.
with B. Virág, Dimension
and randomness in groups acting on rooted trees, J. Amer. Math. Soc. 18
(2005), no. 1, 157-192 ArXiV
We explore the structure of the
p-adic automorphism group Γ(p) of the infinite rooted regular tree. We
determine the asymptotic order of a typical element, answering an old question
of Turan.
We initiate the study of a general dimension theory of groups acting on rooted
trees. We describe the relationship between dimension and other properties of
groups such as solvability, existence of dense free subgroups and the normal
subgroup structure. We show that subgroups of Γ(p) generated by three
random elements are full-dimensional and that there exist finitely generated
subgroups of arbitrary dimension. Specifically, our results solve an open problem
of Shalev and answer a question of Sidki.
with N. Nikolov and B. Szegedy, Congruence subgroup growth of arithmetic
groups in positive characteristic, Duke Math J. (2003), no. 2, 367—383 PDF
We prove a new uniform bound for subgroup growth of a Chevalley group G
over the local ring F[[t]] and also over local pro-p rings of higher Krull
dimension. This is applied to the determination of congruence subgroup growth
of arithmetic groups over global fields of positive characteristic. In
particular we obtain that the subgroup growth of SL_n(F_p[t]) (n > 2) is of type $n^{\log{n}}$. This was one
of the main problems left open by Alex Lubotzky in his
article.
The essential tool for proving the results is the use of graded Lie
algebras. We sharpen Lubotzky`s bounds on subgroup growth via a result on
subspaces of a Chevalley Lie algebra L over a finite field. The proof of this
theorem is by algebraic geometry and can be modified to obtain a lower bound on
the codimension of proper Lie subalgebras of L.
with T. Keleti, Shuffle the plane, Proc. Amer.
Math. Soc. 130 (2002), no. 2, 549-553 PS
We prove that every permutation of the real plane can be obtained as the
composition of a fixed number of simple transformations of the form (x,y) ® (x,y+f(x)) and (x,y) ® (x+g(y),y) where f and g are arbitrary
real functions.
Symmetric groups as products of
Abelian subgroups, Bull. London Math. Soc. 34 (2002), no. 4, 451-456 PS
We prove that the full symmetric group over any infinite set is the product
of finitely many Abelian subgroups. In fact, 289 subgroups suffice.
We also obtain sharp bounds on the minimal number k such that the finite
symmetric group S(n) is the product of k Abelian subgroups. Using this, we
prove that S(n) is the product of $72n^{1/2}(\log n)^{3/2}$ cyclic subgroups.
with A. Lubotzky and L. Pyber, Bounded generation and linear groups,
Int. J. Alg. Comp., 13 (2003), no. 4, 401-413 PS
A group is called boundedly generated (BG) if it is the set-theoretic
product of finitely many cyclic subgroups. We show that a BG group has only
abelian by finite images in positive characteristic representations.
We use this to reprove and generalise Rapinchuk's theorem by showing that a
BG group with the FAb property has only finitely many irreducible
representations in any given dimension over any field. We also give a structure
theorem for the profinite completion of such a group.
On the other hand, we exhibit boundedly generated profinite FAb groups
which do not satisfy this structure theorem.
with L. Pyber, Decomposing finite linear groups into the product of
cyclic groups
Let K be a field. We show that if G is a finite completely reducible subgroup of GL(n,K) then G can be obtained as a product of Cn cyclic subgroups where C is an absolute constant.
Symmetric presentations of
Abelian groups, Proc. Amer. Math. Soc. 131 (2003), no. 1, 17-20 PDF
We characterise the abelianisation of a group, that has a presentation for
which the set of relations is invariant under the full symmetric group acting
on the set of generators. This improves a result of Emerson.