Said Sidki

On the Burnside Problem for Periodic Groups

Narain Gupta 1,. and Said Sidki 2 1 University of Manitoba, Department of Mathematics, Winnipeg, Manitoba R3T 2N2, Canada 2 University of Brasilia, Department of Mathematics, Brasilia, D.F., Brazil Introduction The generalized Burnside problem refers to the question: Are finitely generated periodic groups finite? This was answered in the negative by Golod [1] who proved that, for each prime p, there exists a finitely generated infinite p-group. Golods construction in not, however, direct and is based on his celebrated work with Safarevi~. Recently, Grigor~uk [2] has given a direct and elegant construction of an infinite 2-group which is generated by three elements of order 2. In this paper we give, for each odd prime p, a direct construction of an infinite p-group on two generators, each of order p. Our group is a subgroup of the automorphism group of a regular tree of degree p; and as might be expected, it is residually finite and has infinite exponent. Preliminaries. Let p be an odd prime and let T(0) be the infinite regular tree of degree p with vertex 0, so that through each vertex u of T(0) there are p regular subtrees

Automorphisms of one-rooted trees: Growth, circuit structure, and acyclicity

A natural interpretation of automorphisms of one-rooted trees as output automata permits the application of notions of growth and circuit structure in their study. New classes of groups are introduced corresponding to diverse growth functions and circuit structure. In the context of automorphisms of the binary tree, we discuss the structure of maximal 2-subgroups and the question of existence of free subgroups. Moreover, we construct Burnside 2-groups generated by automorphisms of the binary tree which are finite state, bounded, and acyclic.

The Generation of GL(n, Z) by Finite State Automata

The linear group GL(n, Z) is residually finite by virtue of its action on the (one-rooted) regular 2n-ary coset tree for Zn. In this paper we construct finite state automata which effect this action. This shows that GL(n, Z) is embeddable in the group of finite state automorphisms of the one-rooted regular tree of valency 2n.

A Primitive Ring Associated to a Burnside 3-Group

Let [Gscr ] be a Burnside 3-group constructed by Gupta-Sidki. The group algebra ℤ 3 [[Gscr ]] has a proper ring-quotient which embeds [Gscr ] and is just-infinite and primitive.

THE GROUP OF AUTOMORPHISMS OF A 3-GENERATED 2-GROUP OF INTERMEDIATE GROWTH

The automorphism group of a 3-generated 2-group G of intermediate growth is determined and it is shown that the outer group of automorphisms of G is an elementary abelian 2-group of infinite rank.

Abelian state-closed subgroups of automorphisms of m-ary trees

The group Am of automorphisms of a one-rooted m-ary tree admits a diagonal monomorphism which we denote by x. Let A be an abelian state-closed (or self-similar) subgroup of Am. We prove that the combined diagonal and tree-topological closure Aof A is additively a finitely presented ZmŒŒx�� -module, where Zm is the ring of m-adic integers. Moreover, if Ais torsion-free then it is a finitely generated pro-m group. Furthermore, the group A splits over its torsion subgroup. We study in detail the case where Ais additively a cyclic ZmŒŒx�� -module, and we show that when m is a prime number then Ais conjugate by

ON TORSION-FREE METABELIAN GROUPS WITH COMMUTATOR QUOTIENTS OF PRIME EXPONENT

Let G be a torsion-free metabelian group having for commutator quotient, an elementary abelian p-group of rank k. It is shown that k≥3 for all primes p. Examples of such metabelian torsion-free groups are constructed for all primes p and all ranks k≥3, except for p=2, k=3.

On the conjugacy problem for finite-state automorphisms of regular rooted trees (with an appendix by Raphaël M. Jungers)

We study the conjugacy problem in the automorphism group Aut(T) of a regular rooted tree T and in its subgroup FAut(T) of finite-state automorphisms. We show that under the contracting condition and the finiteness of what we call the orbit-signalizer, two finite-state automorphisms are conjugate in Aut(T) if and only if they are conjugate in FAut(T), and that this problem is decidable. We prove that both these conditions are satisfied by bounded automorphisms and establish that the (simultaneous) conjugacy problem in the group of bounded automata is decidable. Mathematics Subject Classification 2000: 20E08, 20F10

TREE-WREATHING APPLIED TO GENERATION OF GROUPS BY FINITE AUTOMATA

We extend the tree-wreath product of groups introduced by Brunner and the author, as a generalization of the restricted wreath product, thus enlarging the class of groups known to be generated by finite synchronous automata. In particular, we prove that given a countable abelian residually finite 2 -group H and B = B(n,ℤ), a canonical subgroup of finite index in GL(n,ℤ), then the restricted wreath product HwrB can be generated by finite synchronous automata on 0,1. This is obtained by producing a representation of B as a group of automorphisms of the binary tree such that the stabilizer of the infinite sequence of 0s is trivial. The uni-triangular group U = U(n,ℤ) is a subgroup of B(n,ℤ) and so, HwrU also can be generated by finite synchronous automata on 0,1.

Wreath operations in the group of automorphisms of the binary tree

Abstract A new operation called tree-wreathing is defined on groups of automorphisms of the binary tree. Given a countable residually finite 2-group H and a free abelian group K of finite rank r this operation produces uniformly copies of these as automorphism groups of the binary tree such that the group generated by them is an over-group of the restricted wreath product H≀K . Indeed, G contains a normal subgroup N which is an infinite direct sum of copies of the derived group H′ and the quotient group G/N is isomorphic to H≀K . The tree-wreathing construction preserves the properties of solvability, torsion-freeness and of having finite state (i.e., generated by finite automata). A faithful representation of any free metabelian group of finite rank is obtained as a finite-state group of automorphisms of the binary tree.