We exhibit infinite, solvable, virtually abelian groups with a fixed number of generators, having arbitrarily large balls consisting of torsion elements. We also provide a sequence of 3-generator non-virtually nilpotent polycyclic groups of algebraic entropy tending to zero. All these examples are obtained by taking appropriate quotients of finitely presented groups mapping onto the first Grigorchuk group.
We construct the first examples of finitely presented groups with cubic Dehn function containing a finitely generated infinite torsion subgroup. Moreover, we show that any infinite free Burnside group with sufficiently large odd exponent can be embedded as a subgroup of a finitely presented group with cubic Dehn function.
We study $k$-systolic complexes introduced by T. Januszkiewicz and J. Świątkowski, which are simply connected simplicial complexes of simplicial nonpositive curvature. Using techniques of filling diagrams we prove that for $k \geq 7$ the $1$-skeleton of a $k$-systolic complex is Gromov hyperbolic. We give an elementary proof of the so-called Projection Lemma, which implies contractibility of $6$-systolic complexes. We also present a new proof of the fact that an infinite group acting geometrically on a $6$-systolic complex is not torsion.
In this paper we exhibit Morse geodesics, often called "hyperbolic directions", in infinite unbounded torsion groups. The groups studied are lacunary hyperbolic groups and constructed using graded small cancellation conditions. In all previously known examples, Morse geodesics were found in groups which also contained Morse elements, infinite order elements whose cyclic subgroup gives a Morse quasi-geodesic. Our result presents the first example of a group which contains Morse geodesics but no Morse elements. In fact, we show that there is an isometrically embedded $7$-regular tree inside such…
We show that every Coxeter group that is not virtually abelian and for which all labels in the corresponding Coxeter graph are powers of 2 or infinity can be mapped onto uncountably many infinite 2-groups which, in addition, may be chosen to be just-infinite, branch groups of intermediate growth. Also we answer affirmatively a question raised by Wiegold in Kourovka Notebook.
We construct the first examples of finitely presented groups with quadratic Dehn function containing a finitely generated infinite torsion subgroup. These examples are "optimal" in the sense that the Dehn function of any such finitely presented group must be at least quadratic. Moreover, we show that for any $n\geq2^{48}$ such that $n$ is either odd or divisible by $2^9$, any infinite free Burnside group with exponent $n$ is a quasi-isometrically embedded subgroup of a finitely presented group with quadratic Dehn function satisfying the Congruence Extension Property.
We construct first examples of infinite finitely generated residually finite torsion groups with positive rank gradient. In particular, these groups are non-amenable. Some applications to problems about cost and $L^2$-Betti numbers are discussed.
We construct a 2-generator recursively presented group with infinite torsion length. We also explore the construction in the context of solvable and word-hyperbolic groups.
We give an infinite family of torsion-free groups that do not satisfy the unique product property. For these examples, we also show that each group contains arbitrarily large sets whose square has no uniquely represented element.
It is known that every torsion-free abelian group of finite rank has a maximal completely decomposable summand that is unique up to isomorphism. We show that groups of infinite rank need not have maximal completely decomposable summands, but when they do, this summand is unique up to isomorphism.
We show, under mild hypotheses, that if each element of a finitely generated group acting on a $2$-dimensional $\mathrm{CAT}(0)$ complex has a fixed point, then there is a global fixed point. In particular all actions of finitely generated torsion groups on such complexes have global fixed points. The proofs rely on Masur's theorem on periodic trajectories in rational billiards, and Ballmann-Brin's methods for finding closed geodesics in $2$-dimensional locally $\mathrm{CAT}(0)$ complexes. As another ingredient we prove that the image of an immersed loop in a graph of girth $2π$ with length no…
A conjecture due to Zassenhaus asserts that if $\ G$ is a finite group then any torsion unit in $\mathbb{Z}G$ is conjugate in $\mathbb{Q}G$ to an element of $\ G$. We present a weaker form of this conjecture for some infinite groups.
A Grigorchuk-Gupta-Sidki (GGS-)group is a subgroup of the automorphism group of the $p$-adic tree for an odd prime $p$, generated by one rooted automorphism and one directed automorphism. Pervova proved that all torsion GGS-groups do not have maximal subgroups of infinite index. Here we extend the result to non-torsion GGS-groups, which include the weakly regular branch, but not branch, GGS-group.
For any countable torsion subgroup $H$ of an unbounded Abelian group $G$ there is a complete Hausdorff group topology $τ$ such that $H$ is the von Neumann radical of $(G,τ)$. In particular, any unbounded torsion countable Abelian group admits a complete Hausdorff minimally almost periodic (MinAP) group topology. If $G$ is a bounded torsion countably infinite Abelian group, then it admits a MinAP group topology if and only if all its leading Ulm-Kaplansky invariants are infinite. In such a case, a MinAP group topology can be chosen to be complete.
In this article, we state and prove a general criterion which prevents some groups from acting properly on finite-dimensional CAT(0) cube complexes. As an application, we show that, for every non-trivial finite group $F$, the lamplighter group $F \wr \mathbb{F}_2$ over a free group does not act properly on a finite-dimensional CAT(0) cube complex (although it acts properly on a infinite-dimensional CAT(0) cube complex). We also deduce from this general criterion that, roughly speaking, given a group $G$ acting on a CAT(0) cube complex of finite dimension and an infinite torsion subgroup $L \le…
This work examines the commutator structure of some closed subgroups of the wild group of automorphisms of a local field with perfect residue field, a group we call $\Cal J.$ In particular, we establish a new approach to evaluating commutators in $\Cal J$ and using this method investigate the normal subgroup structure of some classes of index subgroups of $\Cal J$ as introduced by Klopsch. We provide new proofs of Fesenko\rq s results that lead to a proof that the torsion free group $T =\{t+\sum_{k\geq 1} a_kt^{qk+1}: a_k \in \Bbb F_p\}$ is hereditarily just infinite, and by extending his work…
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 (2, 2q+1)-torus knot K, we demonstrate that there are infinitely many unknots c such that p-twisting K about c yields a twist family, which consists of hyperbolic knots with generalized torsion whenever |p| > 3. This gives a new infinite class of hyperbolic knots having generalized torsion. In particular, each class contains knots with arbitrarily high genus…
Chung and Jiang showed that, if a one ended group contains an infinite order element, then every continuous cocycle over the full shift on that group, taking values in a discrete group, must be cohomologous to a homomorphism. We show that their conclusion holds for all one ended groups, so that the hypothesis of admitting an infinite order element may be omitted.
It is well-known that a complete Riemannian manifold M which is locally isometric to a symmetric space is covered by a symmetric space. Here we prove that a discrete version of this property (called local to global rigidity) holds for a large class of vertex-transitive graphs, including Cayley graphs of torsion-free lattices in simple Lie groups, and Cayley graph of torsion-free virtually nilpotent groups. By contrast, we exhibit various examples of Cayley graphs of finitely presented groups (e.g. SL(4,Z)) which fail to have this property, answering a question of Benjamini, Ellis, and Georgako…
In a series of papers starting in [Sel01] and culminating in [Sel07], Z. Sela proved that free groups, and more generally torsion-free hyperbolic groups, have a stable first-order theory. The question of the stability of the free product of two arbitrary stable groups has then been raised by E. Jaligot with, seemingly, the reasonable conjecture of a positive answer [Jal08]. The complete proof however will be a grand project of generalization of above papers of Sela.In the meantime, we provide here a very preliminary -or somehow experi- mental- result in the direction of the stability of free p…
We generalize Bonahon's characterization of geometrically infinite torsion-free discrete subgroups of PSL(2, $\mathbb{C}$) to geometrically infinite discrete isometry subgroups in the case of rank 1 symmetric spaces, and, under the assumption of bounded torsion, to the case of negatively pinched Hadamard manifolds. Every such geometrically infinite isometry subgroup $Γ$ has a set of nonconical limit points with cardinality of continuum.
We show that a construction by Aanderaa and Cohen used in their proof of the Higman Embedding Theorem preserves torsion length. We give a new construction showing that every finitely presented group is the quotient of some $C'(1/6)$ finitely presented group by the subgroup generated by its torsion elements. We use these results to show there is a finitely presented group with infinite torsion length which is $C'(1/6)$, and thus word-hyperbolic and virtually torsion-free.
One way to understand the mod p homotopy theory of classifying spaces of finite groups is to compute their BZ/p-cellularization. In the easiest cases this is a classifying space of a finite group (always a finite p-group). If not, we show that it has infinitely many non-trivial homotopy groups. Moreover they are either p-torsion free or else infinitely many of them contain p-torsion. By means of techniques related to fusion systems we exhibit concrete examples where p-torsion appears.
A nontrivial element in a group is a generalized torsion element if some nonempty finite product of its conjugates is the identity. We prove that any generalized torsion element in a free product of torsion-free groups is conjugate to a generalized torsion element in some factor group. This implies that the fundamental group of a compact orientable $3$-manifold $M$ has a generalized torsion element if and only if the fundamental group of some prime factor of $M$ has a generalized torsion element. On the other hand, we demonstrate that there are infinitely many toroidal $3$-manifolds whose fund…
Let R be any ring (with 1), Γa group and RΓthe corresponding group ring. Let Ext_{RΓ}^{*}(M,M) be the cohomology ring associated to the RΓ-module M. Let H be a subgroup of finite index of Γ. The following is a special version of our main Theorem: Assume the profinite completion of Γis torsion free. Then an element ζin Ext_{RΓ}^{*}(M,M) is nilpotent (under Yoneda's product) if and only if its restriction to Ext_{RH}^{*}(M,M)$ is nilpotent. In particular this holds for the Thompson group. There are torsion free groups for which the analogous statement is false.
It is known that a bi-orderable group has no generalized torsion element, but the converse does not hold in general. We conjecture that the converse holds for the fundamental groups of 3-manifolds, and verify the conjecture for non-hyperbolic, geometric 3-manifolds. We also confirm the conjecture for some infinite families of closed hyperbolic 3-manifolds. In the course of the proof, we prove that each standard generator of the Fibonacci group F(2,m) (m>2) is a generalized torsion element.
Boyd (1974) proposed a class of infinite ball packings that are generated by inversions. Later, Maxwell (1983) interpreted Boyd's construction in terms of root systems in Lorentz space. In particular, he showed that the space-like weight vectors correspond to a ball packing if and only if the associated Coxeter graph is of "level $2$." In Maxwell's work, the simple roots form a basis of the representations space of the Coxeter group. In several recent studies, the more general based root system is considered, where the simple roots are only required to be positively independent. In this paper,…
In this note we describe the role of the Schur multiplier in the structure of the $p$-torsion of discrete groups. More concretely, we show how the knowledge of $H_2G$ allows to approximate many groups by colimits of copies of finite $p$-groups. Our examples include interesting families of non-commutative infinite groups, including Burnside groups, certain solvable examples and the first Grigorchuk group. We also provide a counterexample for a conjecture of E. Farjoun.
