Related papers: The Tits alternative for generalized triangle grou…

Let $G$ be a group and $g$ a non-trivial element in $G$. If some non-empty finite product of conjugates of $g$ equals to the trivial element, then $g$ is called a generalized torsion element. To the best of our knowledge, we have no…

In a group, a non-trivial element is called a generalized torsion element if some non-empty finite product of its conjugates equals to the identity. We say that a knot has generalized torsion if its knot group admits such an element. For a…

The proof of the Tits alternative for $Out(F_n)$ is completed. The main tool is a Kolchin type theorem, proved in this paper. It states that a finitely generated subgroup of $Out(F_n)$ consisting of unipotent automorphisms can be conjugated…

We show that for every integer $d$, there is a constant $N(d)$ such that if $K$ is any field and $F$ is a finite subset of $GL_d(K)$, which generates a non amenable subgroup, then $F^{N(d)}$ contains two elements, which freely generate a…

A theorem of Cantat and Urech says that an analog of the classical Tits alternative holds for the group of birational automorphisms of a compact complex Kaehler surface. We established in our previous paper the following Tits-type…

We prove the Tits Alternative for groups acting on $2$-dimensional $\mathrm{CAT}(0)$ complexes with a bound on the order of the cell stabilisers.

We describe new classification results in the theory of generalized quadrangles (= Tits-buildings of rank $2$ and type $\mathbb{B}_2$), more precisely in the (large) sub theory of skew translation generalized quadrangles ("STGQs"). Some of…

Let $G$ be a group with a non-elementary action on a (not necessarily discrete) $\tilde{A}_2$-buildings. We prove that, given a random walk on $G$, isometries in $G$ are strongly regular hyperbolic with high probability. As a consequence,…

A recent article of J.P. Bell, K. Huang, W. Peng and T.J. Tucker establishes an analog of the Tits alternative for semigroups of endomorphisms of the projective line. The proof involves a ping-pong argument on arithmetic height functions.…

The main aim of this paper is to prove $R$-triviality for simple, simply connected algebraic groups with Tits index $E_{8,2}^{78}$ or $E_{7,1}^{78}$, defined over a field $k$ of arbitrary characteristic. Let $G$ be such a group. We prove…

We give a necessary and sufficient condition on a matrix for its centralizer in $\sf{GL}(n,\mathbb{Z})$ to be polycyclic, or equivalently in this case, not to contain a non-abelian free subgroup. We give a simple condition on the matrix…

A triangle group is denoted by $\Delta(p,q,r)$ and has finite presentation $$ \Delta(p,q,r)=\langle x,y | x^p=y^q=(xy)^r=1 \rangle .$$ We examine a method for composition of permutation representations of a triangle group $\Delta(p,q,r)$…

Given a toric affine algebraic variety $X$ and a collection of one-parameter unipotent subgroups $U_1,\ldots,U_s$ of $\mathop{\rm Aut}(X)$ which are normalized by the torus acting on $X$, we show that the group $G$ generated by…

We discuss various types of Tits Alternative for subgroups of graph products of groups, and prove that, under some natural conditions, a graph product of groups satisfies a given form of Tits Alternative if and only if each vertex group…

In [9] Bogdan Nica presented an elementary proof of a result which says that the relative elementary linear group with respect to a square of an ideal of the ring is a subset of the true relative elementary linear group. The original result…

Let $G$ be a group and $R,S,T$ its normal subgroups. There is a natural extension of the concept of commutator subgroup for the case of three subgroups $\|R,S,T\|$ as well as the natural extension of the symmetric product $\|\bf r,\bf s,\bf…

We prove an analog of the Tits alternative for rational functions. In particular, we show that if $S$ is a finitely generated semigroup of rational functions over the complex numbers, then either $S$ has polynomially bounded growth or $S$…

In this paper we study a group G which is the quotient of a free product of groups by the normal closure of a word that is contained in a in a subgroup which has the form of a generalised triangle group. We use known properties of…

Let $G$ be a group, $m\geq2$ and $n\geq1$. We say that $G$ is an $\mathcal{T}(m,n)$-group if for every $m$ subsets $X_1, X_2, \dots, X_m$ of $G$ of cardinality $n$, there exists $i\neq j$ and $x_i \in X_i, x_j \in X_j$ such that…

We prove: (1) The group of multipliers of similitudes of a 12-dimensional anisotropic quadratic form over a field K with trivial discriminant and split Clifford invariant is generated by norms from quadratic extensions E/K such that q_E is…