Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · math/0610946

Basis-conjugating automorphisms of a free group and associated Lie algebras

F R Cohen·J Pakianathan·V V Vershinin·Jie Wu
View original source ↗Read source document ↗Export metadata ↓

Abstract

Let F_n = denote the free group with generators {x_1,...,x_n}. Nielsen and Magnus described generators for the kernel of the canonical epimorphism from the automorphism group of F_n to the general linear group over the integers. In particular among them are the automorphisms chi_{k,i} which conjugate the generator x_k by the generator x_i leaving the x_j fixed for j not k. A computation of the cohomology ring as well as the Lie algebra obtained from the descending central series of the group generated by chi_{k,i} for i<k is given here. Partial results are obtained for the group generated by all chi_{k,i}.

Explore related subjects

math.GRmath.AT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

F R Cohen, J Pakianathan, V V Vershinin, Jie Wu. 2009-04-03. Basis-conjugating automorphisms of a free group and associated Lie algebras. https://doi.org/10.2140/gtm.2008.13.147

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On a Completion of Cohomological Functors Generalising Tate Cohomology I

Tate cohomology has been generalised by several authors using different constructions that have applications in group theory, ring theory and homotopical algebra. Therefore, there is a need for a uniform account that explains why their underlying approaches all lead to the same conclusions. The key notion in such a uniform theory is a specific completion of cohomological functors that is constructed under mild assumptions. This completion takes Tate cohomology to settings where it has never been introduced such as in condensed mathematics. Through the latter, one can define Tate cohomology for any $T1$ topological group.

math.GR↗
arXiv2026

$L^2$-Betti numbers of Dehn fillings

We initiate the study of the $L^2$-Betti numbers of group-theoretic Dehn fillings. For a broad class of virtually special groups $G$, we prove that the $L^2$-Betti numbers of sufficiently deep Dehn fillings $\overline{G}$ are equal to those of $G$. As applications, we verify the Singer Conjecture for certain Einstein manifolds, establish a virtual fibering criterion for $\overline{G}$, obtain bounds on deficiency of $\overline{G}$, and provide new examples of hyperbolic groups with exotic subgroups that arise as Dehn fillings of any cusped arithmetic hyperbolic manifold of dimension at least four.

math.GR↗
arXiv2026

On Zappa-Szép products of two semidihedral groups

Let $n, m \ge 4$. We study the Zappa--Szép products $G = HK$ with $H = \langle x\rangle \rtimes \langle y\rangle \cong SD_{2^n}$ and $K = \langle z\rangle \rtimes \langle w\rangle \cong SD_{2^m}$, together with the fixed generators $x, y, z, w$, and we describe the parameter tuples which encode the mixed commutators of these generators, according to the cores of $\langle x\rangle$ and $\langle z\rangle$ in~$G$. First, when both $\langle x\rangle$ and $\langle z\rangle$ are normal in~$G$, we show that the admissible parameter tuples are exactly the solutions of an explicit system of six polynomial congruences. Second, when the cores $\langle x\rangle_G$ and $\langle z\rangle_G$ are arbitrary subgroups of $\langle x\rangle$ and $\langle z\rangle$, under the simplifying assumption $[x, z] = 1$, and for generating systems for which all mixed commutators lie in $\langle x\rangle\langle z\rangle$, we obtain an analogous classification by twelve congruences together with two order conditions; this is the semidihedral counterpart of the Hu--Yu classification~\cite{HuYu2025} for dihedral groups $D_{2n}$ and $D_{2m}$ with $n, m$ odd. In contrast with that case, we further construct an explicit exact product with both cores non-trivial and $[x, z] \ne 1$, so that the semidihedral setting contains additional families of exact products.

math.GR↗