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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, accepted in 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, accepted in J. Amer. Math. Soc.
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.
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
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
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
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
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.