arXiv subjects

Ihechukwu Chinyere

Publications and source records attributed to Ihechukwu Chinyere.

16 recordsLinked to original sources

arXiv2026

Unipotence of a double commutator with transvections

Let $R$ be a nonzero commutative ring with identity, $n\geq3$, and $σ\in GL_n(R)$. We prove that there exist nonidentity transvections $τ_1,τ_2\in GL_n(R)$ such that $\big[[σ,τ_1],τ_2\big]=I_n$. Hence, Kourovka Notebook Problem 10.46 has a stronger affirmative answer over commutative rings with identity. The proof uses rank-one operators and the existence of a nonzero functional annihilating a standard basis vector and its image under $σ$.

math.GR↗

arXiv2026

Solvable groups defined by redundant cyclic presentations

We study the two-generator one-relator groups defined by $\langle x_0,x_1\mid U(x_0,x_1)U(x_1,x_0)^{-1}\rangle$, and determine exactly which of them fail to contain a free subgroup of rank $2$. We show that every such group is a solvable Baumslag--Solitar group $BS(1,m)$, and that every integer value $m$ can occur. This disproves a conjecture predicting that only $m\in \{-1,0,1\}$ can occur.

math.GR↗

arXiv2026

On the Classification of Perfect Prishchepov Groups

The Prishchepov groups $P(r,n,k,s,q)$ form a broad class of cyclically presented groups. We verify a conjectural characterisation of the perfect groups in this family. We first prove the conjecture for the case $\gcd(n,6)=1$ and then establish further cases beyond this coprimality condition. Consequently, we obtain a classification of perfect Prishchepov groups in a broad range of parameters.

math.GR↗

arXiv2026

Groups with special presentations and star-graph $K_{3,3}$

We consider a question of Edjvet and Vdovina concerning which groups defined by special presentations are large. For each integer $n \ge 3$, we construct an $n$-generator one-relator presentation whose star graph is the complete bipartite graph $K_{n,n}$; the resulting groups are large and hyperbolic. We also classify concise special presentations with star graph $K_{3,3}$, showing that they are one-relator presentations and that, up to Tietze equivalence, there are exactly twelve that define torsion-free groups. The torsion cases arise precisely as positive powers of the relators in the torsion-free cases, and define pairwise non-isomorphic groups that remain large and hyperbolic.

math.GR↗

arXiv2026

Proof of the Noferini-Williams conjecture for Gilbert-Howie groups

The Gilbert-Howie groups $H(n,m)$ form a notable subclass within the broader family of Fibonacci-type cyclically presented groups $G_n(m,k)$. Noferini and Williams conjectured that the abelianization $H(n,m)^{ab}$ is torsion-free with $\mathbb{Z}$-rank $2$ if and only if $n\equiv 0\pmod{6}$ and $m\equiv 2\pmod{n}$. We confirm this conjecture by proving that $\mathrm{Res}(F,G)=\pm1$, where $F=(1+t^m-t)/Φ_6$ and $G=(t^n-1)/Φ_6$, with $Φ_6$ denoting the sixth cyclotomic polynomial. The proof uses a minimality argument, reducing the general problem to three cases: $m=2+n/3$, $m=2+n/2$, and $m=2+2n/3$. These cases are handled using polynomial resultant analysis and field-theoretic methods. As a consequence, we complete the classification of all $G_n(m,k)$ that arise as labelled oriented graph groups.

math.GR↗

arXiv2026

On the non-existence of finite groups with certain normal subgroups

Problem 20.21 of Mazurov and Khukhro (Unsolved Problems in Group Theory: The Kourovka Notebook, 20th Issue, 2022), contributed by M.~Conder and attributed to G.~Verret, asks whether there exists a finite group $G$ with two normal subgroups $K$ and $L$ of index $12$ such that $K \cong L$, but with non-isomorphic quotients $G/K \cong C_{12}$ and $G/L \cong A_4.$ We prove that no such finite group exists.

math.GR↗

arXiv2025

All hyperbolic cyclically presented groups with positive length three relators

We consider the cyclically presented groups defined by cyclic presentations with $2m$ generators $x_i$ whose relators are the $2m$ positive length three relators $x_ix_{i+1}x_{i+m-1}$. We show that they are hyperbolic if and only if $m\in \{1,2,3,6,9\}$. This completes the classification of the hyperbolic cyclically presented groups with positive length three relators.

math.GR↗

arXiv2022

Fractional Fibonacci groups with an odd number of generators

The Fibonacci groups $F(n)$ are known to exhibit significantly different behaviour depending on the parity of $n$. We extend known results for $F(n)$ for odd $n$ to the family of Fractional Fibonacci groups $F^{k/l}(n)$. We show that for odd $n$ the group $F^{k/l}(n)$ is not the fundamental group of an orientable hyperbolic 3-orbifold of finite volume. We obtain results concerning the existence of torsion in the groups $F^{k/l}(n)$ (where $n$ is odd) paying particular attention to the groups $F^k(n)$ and $F^{k/l}(3)$, and observe consequences concerning asphericity of relative presentations of their shift extensions. We show that if $F^{k}(n)$ (where $n$ is odd) and $F^{k/l}(3)$ are non-cyclic 3-manifold groups then they are isomorphic to the direct product of the quaternion group $Q_8$ and a finite cyclic group.

math.GR↗

arXiv2021

Redundant relators in cyclic presentations of groups

A cyclic presentation of a group is a presentation with an equal number of generators and relators that admits a particular cyclic symmetry. We characterise the orientable, non-orientable, and redundant cyclic presentations and obtain concise refinements of these presentations. We show that the Tits alternative holds for the class of groups defined by redundant cyclic presentations and that if the number of generators of the cyclic presentation is greater than two then the corresponding group is large. Generalizing and extending earlier results of the authors we describe the star graphs of orientable and non-orientable cyclic presentations and classify the cyclic presentations whose star graph components are pairwise isomorphic incidence graphs of generalized polygons, thus classifying the so-called $(m,k,ν)$-special cyclic presentations.

math.GR↗

arXiv2021

Hyperbolicity of T(6) Cyclically Presented Groups

We consider groups defined by cyclic presentations where the defining word has length three and the cyclic presentation satisfies the T(6) small cancellation condition. We classify when these groups are hyperbolic. When combined with known results, this completely classifies the hyperbolic T(6) cyclically presented groups.

math.GR↗

arXiv2021

Perfect Prishchepov groups

We study cyclically presented groups of type $\mathfrak{F}$ to determine when they are perfect. It turns out that to do so, it is enough to consider the Prishchepov groups, so modulo a certain conjecture, we classify the perfect Prishchepov groups $P(r,n,k,s,q)$ in terms of the defining integer parameters $r,n,k,s,q$. In particular, we obtain a classification of the perfect Campbell and Robertson's Fibonacci-type groups $H(r,n,s)$, thereby proving a conjecture of Williams, and yielding a complete classification of the groups $H(r,n,s)$ that are connected Labelled Oriented Graph groups.

math.GR↗

arXiv2021

Generalized polygons and star graphs of cyclic presentations of groups

Groups defined by presentations for which the components of the corresponding star graph are the incidence graphs of generalized polygons are of interest as they are small cancellation groups that - via results of Edjvet and Vdovina - are fundamental groups of polyhedra with the generalized polygons as links and so act on Euclidean or hyperbolic buildings; in the hyperbolic case the groups are SQ-universal. A cyclic presentation of a group is a presentation with an equal number of generators and relators that admits a particular cyclic symmetry. We obtain a classification of the non-redundant cyclic presentations where the components of the corresponding star graph are generalized polygons. The classification reveals that both connected and disconnected star graphs are possible and that only generalized triangles (i.e. incidence graphs of projective planes) and regular complete bipartite graphs arise as the components. We list the presentations that arise in the Euclidean case and show that at most two of the corresponding groups are not SQ-universal (one of which is not SQ-universal, the other is unresolved). We obtain results that show that many of the SQ-universal groups are large.

math.GR↗

arXiv2021

Hyperbolic groups of Fibonacci type and T(5) cyclically presented groups

Building on previous results concerning hyperbolicity of groups of Fibonacci type, we give an almost complete classification of the (non-elementary) hyperbolic groups within this class. We are unable to determine the hyperbolicity status of precisely two groups, namely the Gilbert-Howie groups H(9,4), H(9,7). We show that if H(9,4) is torsion-free then it is not hyperbolic. We consider the class of T(5) cyclically presented groups and classify the (non-elementary) hyperbolic groups and show that the Tits alternative holds.

math.GR↗

arXiv2018

Structure of words with short 2-length in a free product of groups

Howie and Duncan observed that a word in a free product with length at least two and which is not a proper power can be decomposed as a product of two cyclic subwords each of which is uniquely positioned. Using this property, they proved various important results about one-relator product of groups. In this paper, we show that similar results hold in a more general setting where we allow elements of order two.

math.GR↗

arXiv2015

Non-triviality of some one-relator products of three groups

In this paper we study a group G which is the quotient of a free product of three non-trivial groups by the normal closure of a single element. In particular we show that if the relator has length at most eight, then G is non-trivial. In the case where the factors are cyclic, we prove the stronger result that at least one of the factors embeds in G.

math.GR↗

arXiv2015

On One-relator products induced by generalised triangle groups

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 generalised triangle groups, together with detailed analyses of pictures and of words in free monoids, to prove a number of results such as a Freiheitssatz and the existence of Mayer-Vietoris sequences for such groups under suitable hypotheses. The hypotheses are weaker than those in an earlier article of Howie and Shwartz, yielding generalisations in two directions of the results in that article.

math.GR↗