Related papers: Commutators and squares in free groups

We establish a general criterion for the finite presentability of subdirect products of groups and use this to characterize finitely presented residually free groups. We prove that, for all $n\in\mathbb{N}$, a residually free group is of…

The main result of this paper is an explicit construction of the free commutative skew brace -- that is, a skew brace whose circle group is commutative -- on an arbitrary generating set $X$. We embed this object into a set of rational…

Let $R$ be a finite commutative local principal ring, and let $H(R)$ denote the corresponding quaternion ring. We show that an element of $H(R)$ is a product of idempotents if and only if it can be expressed as a product of two idempotents.…

We apply the free product construction to various local algebras in algebraic quantum field theory. If we take the free product of infinitely many identical half-sided modular inclusions with ergodic canonical endomorphism, we obtain a…

The theory of bounded cohomology of groups has many applications. A key open problem is to compute the full bounded cohomology $H_b^n(F, R)$ of a non-abelian free group $F$ with trivial real coefficients. It is known that $H_b^n(F,R)$ is…

In \cite{jpsf} we constructed pairs of units $u,v$ in $\Z$-orders of a quaternion algebra over $\Q (\sqrt{-d})$, $d \equiv 7 \pmod 8$ positive and square free, such that $< u^ n,v^n>$ is free for some $n\in \mathbb{N}$. Here we extend this…

If $G_1$ and $G_2$ are finite groups with periodic Tate cohomology, then $G_1\times G_2$ acts freely and smoothly on some product $S^n \times S^n$.

If F is a finitely generated free group and \phi is an automorphism of F then the mapping torus of \phi admits a quadratic isoperimetric inequality. This is the third and final paper in a series proving this theorem. The first two were…

In this note, we first work out some `bare hands' computations of the most elementary possible free products involving $\mathbb{C}^2 ~(=\mathbb{C} \oplus \mathbb{C} $) and $M_2 ~(= M_2(\mathbb{C}))$. Using these, we identify all free…

We consider amalgamated free product II$_1$ factors $M = M_1 *_B M_2 *_B ...$ and use ``deformation/rigidity'' and ``intertwining'' techniques to prove that any relatively rigid von Neumann subalgebra $Q\subset M$ can be intertwined into…

The group of automorphisms of the free group on two generators is known to act geometrically, in an essentially unique way, on a 2-dimensional CAT(0) space X. We prove that X contains precisely two Hamiltonian surfaces. By this we mean a…

Let $F$ be a free group of finite rank. We say that the monomorphism problem in $F$ is decidable if for any two elements $u$ and $v$ in $F$, there is an algorithm that determines whether there exists a monomorphism of $F$ that sends $u$ to…

Thompson's theorem stated that a finite group $G$ is solvable if and only if every $2$-generated subgroup of $G$ is solvable. In this paper, we prove some new criteria for both solvability and nilpotency of a finite group using certain…

Let $G=A \ast B$ be a free product of freely indecomposable groups. We explicitly construct quasimorphisms on $G$ which are invariant with respect to all automorphisms of $G$. We also prove that the space of such quasimorphisms is…

We investigate free product structures in R. Thompson's group V, primarily by studying the topological dynamics associated with V's action on the Cantor Set. We show that the class of free products which can be embedded into V includes the…

Two finite groups $L_1$ and $L_2$ are compatible if there exists a finite group $G$ with isomorphic normal subgroups $N_1$ and $N_2$ such that $L_1\cong G/N_1$ and $L_2\cong G/N_2$. We prove a new sufficient condition for two groups to be…

Each element of the commutator subgroup of a group can be represented as a product of commutators. The minimal number of factors in such a product is called the commutator length of the element. The commutator length of a group is defined…

Every hyperbolic group acts continuously on its Gromov boundary. One can form the corresponding cross-product C*-algebra A. We show that there always exists a canonical Poincare duality map from the K-theory of A to the K-homology of A. We…

A finitely generated group G is termed parafree if it is residually nilpotent and it has the same isomorphism types of nilpotent quotients as some free group. The two main results of this MSc. Thesis characterise the parafreeness of two…

We show that on an arbitrary finitely generated non virtually solvable linear group, any two independent random walks will eventually generate a free subgroup. In fact, this will hold for an exponential number of independent random walks.