Related papers: Solving the isomorphism problems for two families …

Let $K$ be a field and $n\geq 3$. Let $E_n(K)\leq H\leq GL_n(K)$ be an intermediate group and $C$ a noncentral $H$-class. Define $m(C)$ as the minimal positive integer $m$ such that $\exists i_1,\dots,i_m\in\{\pm 1\}$ such that the product…

Let i be a homomorphism of the multiplicative group into a connected reductive algebraic group over C. Let G^i be the centralizer of the image i. Let LG be the Lie algebra of G and let L_nG (n integer) be the summands in the direct sum…

In this paper, we construct a mixed-base number system over the generalized symmetric group $G(m,1,n)$, which is a complex reflection group with a root system of type $B_n^{(m)}$. We also establish one-to-one correspondence between all…

For any positive integer $n$, we exhibit a cofinite subgroup $\Gamma_n$ of the mapping class group of a surface of genus at most two such that $\Gamma_n$ admits an epimorphism onto a free group of rank $n$. We conclude that…

This paper is a continuation of two previous papers in MSJ Memoirs, Vol.\,29 (Math. Soc. Japan, 2013) with the same title and numbered as I and II. Based on the hereditary property given there, from mother groups $G(m,1,n)$, the generalized…

Recently, John Franks and Michael Handel proved that, for $g\geq 3$ and $n\leq 2g-4$, every homomorphism from the mapping class group of an orientable surface of genus $g$ to $\GL (n,\C)$ is trivial. We extend this result to $n\leq 2g-1$,…

We exhibit examples of finitely presented subgroups $P$ of direct products of hyperbolic groups for which there is no algorithm that detects whether a finitely presented group has a quotient isomorphic to $P$. For any torsion-free, linear,…

Given an $N$-dimensional compact manifold $M$ and a field $\bk$, F. Cohen and L. Taylor have constructed a spectral sequence, $\cE(M,n,\bk)$, converging to the cohomology of the space of ordered configurations of $n$ points in $M$. The…

Let G be a simple, simply connected algebraic group defined over an algebraically closed field k of positive characteristic p. Let \sigma:G->G be a strict endomorphism (i. e., the subgroup G(\sigma) of \sigma-fixed points is finite). Also,…

We study the algebra of conformal endomorphisms $\Cend^{G,G}_n$ of a finitely generated free module $M_n$ over the coordinate Hopf algebra $H$ of a linear algebraic group $G$. It is shown that a conformal subalgebra of $\Cend_n$ acting…

Let m and n be any integers with n>m>=2. Using just the entropy function it is possible to define a partial order on S_mn (the symmetric group on mn letters) modulo a subgroup isomorphic to S_m x S_n. We explore this partial order in the…

The study of homomorphisms of $(n,m)$-graphs, that is, adjacency preserving vertex mappings of graphs with $n$ types of arcs and $m$ types of edges was initiated by Ne\v{s}et\v{r}il and Raspaud in 2000. Later, some attempts were made to…

We prove a formula for the ${\mathbb S}_n$-equivariant Euler characteristic of the moduli space of graphs $\mathcal{MG}_{g,n}$. Moreover, we prove that the rational ${\mathbb S}_n$-invariant cohomology of $\mathcal{MG}_{g,n}$ stabilizes for…

We present a new algorithm that, given two matrices in $GL(n,Q)$, decides if they are conjugate in $GL(n,Z)$ and, if so, determines a conjugating matrix. We also give an algorithm to construct a generating set for the centraliser in…

The invariant subalgebra H^+ of the Heisenberg vertex algebra H under its automorphism group Z/2Z was shown by Dong-Nagatomo to be a W-algebra of type W(2,4). Similarly, the rank n Heisenberg vertex algebra H(n) has the orthogonal group…

Two graphs are homomorphism indistinguishable over a graph class $\mathcal{F}$, denoted by $G \equiv_{\mathcal{F}} H$, if $\operatorname{hom}(F,G) = \operatorname{hom}(F,H)$ for all $F \in \mathcal{F}$ where $\operatorname{hom}(F,G)$…

Recently, I. J. Leary and A. Minasyan studied the class of groups $G(A,L)$ defined as commensurating HNN-extensions of $\mathbb{Z}^n$. This class, containing the class of Baumslag-Solitar groups, also includes other groups with curious…

Let $\Gamma_{g,1}^m$ be the mapping class group of the orientable surface $\Sigma_{g,1}^m$ of genus $g$ with one parametrised boundary curve and $m$ permutable punctures; when $m=0$ we omit it from the notation. Let…

If Gamma is a nonuniform, irreducible lattice in a semisimple Lie group whose real rank is greater than 1, we show Gamma contains a subgroup that is isomorphic to a nonuniform, irreducible lattice in either SL(3,R), SL(3,C), or a direct…

We investigate the homogeneous $2$-local representations of the twin group $T_n$ for all integers $n\geqslant 2$. A complete classification is obtained, yielding three distinct families of representations. We show that each of these…