-systems of finite simple groups
Tarski's problem about the elementary theory of free groups has a positive solution.
Term functions on non-Abelian groups of order
Test elements and test rank of a free metabelian group.
Test elements and the retract theorem in hyperbolic groups.
The 27-dimensional module for E6. I.
The absolute Galois group of a pseudo p-adically closed field.
The absolute Galois group of a pseudo real closed field
The accessibility of finitely presented groups.
The amalgamation basis of the category of compact groups.
The arithmetic of curves defined by iteration
We show how the size of the Galois groups of iterates of a quadratic polynomial f can be parametrized by certain rational points on the curves Cₙ: y² = fⁿ(x) and their quadratic twists (here fⁿ denotes the nth iterate of f). To that end, we study the arithmetic of such curves over global and finite fields, translating key problems in the arithmetic of polynomial iteration into a geometric framework. This point of view has several dynamical applications. For instance, we establish a maximality theorem...
The asymptotic dimension of the first Grigorchuk group is infinity.
We describe a sufficient condition for a finitely generated group to have infinite asymptotic dimension. As an application, we conclude that the first Grigorchuk group has infinite asymptotic dimension.
The automorphism group of HNN extensions with finite base group
The Automorphism Group of the Free Algebra of Rank Two
The theorem of Czerniakiewicz and Makar-Limanov, that all the automorphisms of a free algebra of rank two are tame is proved here by showing that the group of these automorphisms is the free product of two groups (amalgamating their intersection), the group of all affine automorphisms and the group of all triangular automorphisms. The method consists in finding a bipolar structure. As a consequence every finite subgroup of automorphisms (in characteristic zero) is shown to be conjugate to a group...
The automorphisms of the orthogonal groups ... n (V), n ... 5.
The Baer-Invariant and the Direct Limit.
The Banach-Tarski paradox for the hyperbolic plane (II)
The second author found a gap in the proof of the main theorem in [J. Mycielski, Fund. Math. 132 (1989), 143-149]. Here we fill that gap and add some remarks about the geometry of the hyperbolic plane ℍ².
The based ring of the lowest two-sided cell of an affine Weyl group. II
The Birkhoff-Neumann Embedding of Relatively Free Groups
The categories of free metabelian groups and Lie algebras