arXiv subjects

C. E. Kofinas

Publications and source records attributed to C. E. Kofinas.

8 recordsLinked to original sources

arXiv2026

Presentations of Lower-Triangular Subgroups of $\operatorname{Aut}(F_n)$

Let $F_n$ be the free group of rank $n\geq2$ with basis $x_1,\ldots,x_n$. For $1\leq j<i\leq n$, let $d_{i,j}$ and $e_{i,j}$ be the automorphisms of $F_n$ defined by $d_{i,j}(x_i)=x_ix_j$ and $e_{i,j}(x_i)=x_jx_i$, respectively, and fixing the remaining free generators. Write $D_n=\langle d_{i,j}\mid 1\leq j<i\leq n\rangle$ and $A_n^+=\langle d_{i,j},e_{i,j}\mid 1\leq j<i\leq n\rangle$. We prove that $D_n$ admits a presentation on the generators $d_{r+1,r}$, $1\leq r\leq n-1$, with three families of relations given by commutators of weights two, three, and four, respectively. We extend this to a presentation of $A_n^+$ on the generators $d_{r+1,r}$ and $e_{r+1,r}$, $1\leq r\leq n-1$. These generating sets have minimum cardinality. Moreover, the presentation of $A_n^+$ may be chosen so that every defining relator is a single commutator.

math.GR↗

arXiv2026

The Lower Central Series of Right Lower-Triangular Nielsen Automorphism Groups

Let $F_n$ be a free group of rank $n\geq3$, freely generated by $x_1,\ldots,x_n$. For $1\leq j<i\leq n$, let $d_{i,j}$ denote the Nielsen automorphism of $F_n$ defined by $d_{i,j}(x_i)=x_ix_j$ and $d_{i,j}(x_k)=x_k$ for $k\neq i$, and let $D_n=\langle d_{i,j}\mid 1\leq j<i\leq n\rangle$. We determine the lower central series of $D_n$. We first prove that, for $1\leq r<i\leq n$, $d_{i,r}\in γ_{i-r}(D_n)\setminusγ_{i-r+1}(D_n)$. For each $i=2,\ldots,n$, this calculation leads to a filtration $\{W_{i,c}\}_{c\geq1}$ of $U_i=\langle d_{i,1},\ldots,d_{i,i-1}\rangle$. For every $c\geq1$, we obtain an explicit iterated semidirect-product decomposition of $γ_c(D_n)$ in terms of the subgroups $W_{i,c}$, and prove that $U_i\capγ_c(D_n)=W_{i,c}$ for $i=2,\ldots,n$. The construction also gives an explicit basis for each quotient $γ_c(D_n)/γ_{c+1}(D_n)$ in terms of basic commutators and determines the exact lower-central depth of every such commutator. Consequently, $D_n$ is a Magnus group.

math.GR↗

arXiv2026

Tame Automorphisms of Free Nilpotent Lie Algebras of Countably Infinite Rank

Let $L_{\infty}$ be a free Lie algebra of countably infinite rank over a field of characteristic $0$ and let $L_{\infty, c}$ be the free nilpotent Lie algebra of countably infinite rank and class $c$. We prove that every automorphism of $L_{\infty, c}$ is induced by an automorphism of $L_{\infty}$.

math.RA↗

arXiv2026

Automorphisms of a Free Centre-by-Centre-by-Metabelian Group of Rank 3

Let $F_{3}$ be the free group of rank $3$ and let $G_{3} = F_{3}/[F_{3}^{\prime\prime}, F_{3}, F_{3}]$, that is, $G_{3}$ is a free centre-by-centre-by-metabelian group of rank $3$. We show that ${\rm Aut}(G_{3})$ contains a proper finitely generated subgroup that is dense with respect to the formal power series topology.

math.GR↗

arXiv2022

On automorphisms of certain free nilpotent-by-abelian Lie algebras

For a positive integer $n$, with $n \geq 4$, let $R_{n}$ be a free (nilpotent of class 2)-by-abelian and abelian-by-(nilpotent of class 2) Lie algebra of rank $n$. We show that the subgroup of Aut$(R_{n})$ generated by the tame automorphisms and a countably infinite set of explicitly given automorphisms of $R_{n}$ is dense in Aut$(R_{n})$ with respect to the formal power series topology.

math.GR↗

arXiv2022

Baumslag-Solitar groups and residual nilpotence

For a Baumslag-Solitar group $G$ we calculate the intersection $γ_w(G)$ of all terms of the lower central sequence of $G$.Using this we are able to show that $[γ_w(G),G]=γ_w(G)$ thus answering a question of Bardakov and Neschadim. Finally we show that the quotient groups $γ_c(G)/γ_{c+1}(G)$ of the lower central series of $G$ are finite.

math.GR↗

arXiv2020

Quotient groups of IA-automorphisms of free metabelian groups

For a positive integer $n$, with $n \geq 2$, let $M_n$ be a free metabelian group of rank $n$. For $c \in \mathbb{N}$, let $γ_c(M_n)$ be the $c$-th term of the lower central series of $M_n$. For $c \geq 2$, let ${\rm I}_{c}{\rm A}(M_n)$ be the subgroup of ${\rm Aut}(M_{n})$ consisting of all automorphisms inducing the identity mapping on $M_n/γ_c(M_n)$. In this paper, we study the quotient groups ${\cal L}^{c}({\rm IA}(M_{n})) = {\rm I}_{c}{\rm A}(M_n)/{\rm I}_{c+1}{\rm A}(M_n)$ for all $n$ and $c$. For $c \geq 2$, we show $γ_{c}({\rm IA}(M_{2})) = {\rm I}_{c+1}{\rm A}(M_{2}))$. For $n = 3$, we show $γ_{3}({\rm IA}(M_{3})) \neq {\rm I}_{4}{\rm A}(M_{3})$ and so, the Andreadakis' conjecture (for a free metabelian group) is not valid for $n = 3$ and $c = 3$. For $n \geq 4$ and $c \geq 3$, we prove that ${\cal L}^{c}({\rm IA}(M_{n})) = γ_{c-1}({\rm IA}(M_{n})){\rm I}_{c+1}{\rm A}(M_{n})/{\rm I}_{c+1}{\rm A}(M_{n})$.

math.GR↗

arXiv2019

IA-automorphisms and Lie Algebras related to the McCool group

In the present work we investigate a subgroup $I_n$ of the McCool group $M_n$. We show that $I_n$ has solvable conjugacy problem. Next, we investigate its Lie Algebra gr($I_n$) and we find a presentation for it. Finally we show that gr($I_n$) is naturally embedded into the Andreadakis-Johnson Lie Algebra of the IA automorphisms of the free group $F_n$.

math.RA↗