Narrow Results

In this paper, we study Helly graphs of finite combinatorial dimension, i.e. whose injective hull is finite-dimensional. We describe very simple fine simplicial subdivisions of the injective hull of a Helly graph, following work of Lang. We also give a very explicit simplicial model of the injective hull of a Helly graph, in terms of cliques which are intersections of balls.

We use these subdivisions to prove that any automorphism of a Helly graph with finite combinatorial dimension is either elliptic or hyperbolic. Moreover, every such hyperbolic automorphism has an axis in an appropriate Helly subdivision, and its translation length is rational with uniformly bounded denominator.

Let $n\ge 2$, $F$ a field, and $F^n$ the $n$-dimensional vector space over $F$. In this paper, we introduce an undirected simple graph, called the single zero component graph $\mathrm{\Gamma }(F^n)$ of $F^n$, and determine the minimum independent domination sets as well as automorphisms of $\mathrm{\Gamma }(F^n)$.

In interconnection network topologies, the $n$-dimensional star graph ${St}_n$ has $n!$ vertices corresponding to permutations $a_{\rho (1)},\dots ,a_{\rho (n)}$ of $n$ symbols $a_1,\dots ,a_n$ and edges which exchange the positions of the first symbol $a_{\rho (1)}$ with any one of the other symbols. The star graph compares favorably with the familiar $n$-cube on degree, diameter and a number of other parameters. A desirable property which has not been fully evaluated in star graphs is the presence of multiple edge-disjoint Hamilton cycles which are important for fault-tolerance. The only known method for producing multiple edge-disjoint Hamilton cycles in ${St}_n$ has been to label the edges in a certain way and then take images of a known base 2-labelled Hamilton cycle under different automorphisms that map labels consistently. However, optimal bounds for producing edge-disjoint Hamilton cycles in this way, and whether Hamilton decompositions can be produced, are not known for any ${St}_n$ other than for the case of ${St}_5$ which does provide a Hamilton decomposition. In this paper we show that, for all n, not more than $\phi (n)/2$, where $\phi$ is Euler’s totient function, edge-disjoint Hamilton cycles can be produced by such automorphisms. Thus, for non-prime $n$, a Hamilton decomposition cannot be produced. We show that the $\phi (n)/2$ upper bound can be achieved for all even $n$. In particular, if $n$ is a power of 2, ${St}_n$ has a Hamilton decomposable spanning subgraph comprising more than half of the edges of ${St}_n$. Our results produce a better than twofold improvement on the known bounds for any kind of edge-disjoint Hamilton cycles in $n$-dimensional star graphs for general $n$.

We study topological properties of attracting sets for automorphisms of ℂ^k. Our main result is that a generic volume preserving automorphism has a hyperbolic fixed point with a dense stable manifold. On the other hand, we show that an attracting set can only contain a neighborhood of the fixed point if it is an attracting fixed point. We will see that the latter does not hold in the non-autonomous setting.

The structure theory of a class of generalized loop super-Virasoro algebras is studied in this paper. In particular, the derivation algebras, the automorphism groups and the second cohomology groups of generalized loop centerless supper-Virasoro algebras are determined completely.

Given a Stein manifold with the density property, we show that under a suitable topological condition it is possible to prescribe derivatives at a finite number of points to automorphisms depending holomorphically on a Stein parameter. This is an Oka property of the manifold and is related to its holomorphic flexibility.

We prove that the automorphism group of a general complete intersection $X$ in ${ℂ\mathbb{P}}^n$ is trivial with a few well-understood exceptions. We also prove that the automorphism group of a complete intersection $X$ acts on the cohomology of $X$ faithfully with a few well-understood exceptions.

In this paper, we first show that the monoid of separable surjective self-morphisms of a variety of Ueno type coincides with the group of automorphisms. We also give an explicit description of the automorphism group. As applications, we confirm Kawaguchi–Silverman conjecture for automorphisms of a variety of Ueno type and some Calabi–Yau three-fold, defined over $\overline{\mathbb{Q}}$.

Let F(x,y) be a relatively free algebra of rank 2 in some variety of algebras over a field K of characteristic 0. In this paper we consider the problem whether p(x,y) ∈ F(x,y) is a primitive element (i.e. an automorphic image of x): (i) If F(x,y)/(p(x,y)) ≅ F(z), the relatively free algebra of rank 1 (ii) If p(f,g) is primitive for some injective endomorphism (f,g) of F(x,y) (iii) If p(x,y) is primitive in a relatively free algebra of larger rank. These problems have positive solutions for polynomial algebras in two variables. We give the complete answer for the free metabelian associative and Lie algebras and some partial results for free associative algebras.

For positive integers n and c, with n≥2, let G_n,c be a relatively free group of rank n in the variety N₂A∧AN₂∧N_c. It is shown that there exists an explicitly described finite subset Ω of IA-automorphisms of G_n,c such that the cardinality of Ω is independent upon n and c and the subgroup of the automorphism group Aut(G_n,c) of G_n,c generated by the tame automorphisms and Ω has finite index in Aut(G_n,c). This is a simpler result than one given in [12, Theorem 1(I)]. Let L(G_n,c) be the associated Lie ring of G_n,c and K be a field of characteristic zero. The method developed in the proof of the aforementioned result is applied in order to find an explicitly described finite subset Ω_L of the IA-automorphism group of K⊗L(G_n,c) such that the automorphism group of K⊗L(G_n,c) is generated by GL(n,K) and Ω_L. In particular, for n≥3, the cardinality of Ω_L is independent upon n and c.

Let U be a universal algebra. An automorphism α of the endomorphism semigroup of U defined by α(φ) = sφs^-1 for a bijection s : U → U is called a quasi-inner automorphism. We characterize bijections on U defining such automorphisms. For this purpose, we introduce the notion of a pre-automorphism of U. In the case when U is a free universal algebra, the pre-automorphisms are precisely the well-known weak automorphisms of U. We also provide different characterizations of quasi-inner automorphisms of endomorphism semigroups of free universal algebras and reveal their structure. We apply obtained results for describing the structure of groups of automorphisms of categories of free universal algebras, isomorphisms between semigroups of endomorphisms of free universal algebras, automorphism groups of endomorphism semigroups of free Lie algebras etc.

We show that the following problems are decidable in a rank 2 free group F₂: Does a given finitely generated subgroup H contain primitive elements? And does H meet the orbit of a given word u under the action of G, the group of automorphisms of F₂? Moreover, decidability subsists if we allow H to be a rational subset of F₂, or alternatively if we restrict G to be a rational subset of the set of invertible substitutions (a.k.a. positive automorphisms). In higher rank, the following weaker problem is decidable: given a finitely generated subgroup H, a word u and an integer k, does H contain the image of u by some k-almost bounded automorphism? An automorphism is k-almost bounded if at most one of the letters has an image of length greater than k.

We develop a theory of generalized presentations of groups and give a generalized presentation of the automorphism group of the free group of countable rank, Aut(F_ω).

Let X be a connected partially ordered set and let K be a field of characteristic different from 2. We present necessary and sufficient conditions for two involutions on the finitary incidence algebra of X over K, FI(X), to be equivalent in the case when every multiplicative automorphism of FI(X) is inner. To get the classification of involutions we extend the concept of multiplicative automorphism to finitary incidence algebras and prove the Decomposition Theorem of involutions of [Anti-automorphisms and involutions on (finitary) incidence algebras, Linear Multilinear Algebra60 (2012) 181–188] for finitary incidence algebras.

Let ${\mathrm{\Gamma }}_d(q)$ denote the group whose Cayley graph with respect to a particular generating set is the Diestel–Leader graph $D{L}_d(q)$, as described by Bartholdi, Neuhauser and Woess. We compute both $\mathrm{\text{Aut}}({\mathrm{\Gamma }}_d(q))$ and $\mathrm{\text{Out}}({\mathrm{\Gamma }}_d(q))$ for $d\ge 2$, and apply our results to count twisted conjugacy classes in these groups when $d\ge 3$. Specifically, we show that when $d\ge 3$, the groups ${\mathrm{\Gamma }}_d(q)$ have property $R_{\infty }$, that is, every automorphism has an infinite number of twisted conjugacy classes. In contrast, when $d=2$ the lamplighter groups ${\mathrm{\Gamma }}_2(q)=L_q={\mathbb{Z}}_q\wr \mathbb{Z}$ have property $R_{\infty }$ if and only if $(q,6)\ne 1$.

We give a sharpening of a result of Bryant and Drensky [R. M. Bryant and V. Drensky, Dense subgroups of the automorphism groups of free algebras, Canad. J. Math.45(6) (1993) 1135–1154] for the automorphism group $\mathrm{\text{Aut}}(M_n)$ of a free metabelian Lie algebra $M_n$, with $n\ge 3$. In particular, we prove that the subgroup of $\mathrm{\text{Aut}}(M_3)$ generated by ${\mathrm{\text{GL}}}_3(K)$ and two more IA-automorphisms is dense in $\mathrm{\text{Aut}}(M_3)$ and, for $n\ge 4$, the subgroup generated by ${\mathrm{\text{GL}}}_n(K)$ and one more IA-automorphism is dense in $\mathrm{\text{Aut}}(M_n)$.

We study topological properties of Ind-groups $\mathrm{\text{Aut}}(K[x_1,\dots ,x_n])$ and $\mathrm{\text{Aut}}(K〈x_1,\dots ,x_n〉)$ of automorphisms of polynomial and free associative algebras via Ind-schemes, toric varieties, approximations, and singularities. We obtain a number of properties of $\mathrm{\text{Aut}}(\mathrm{\text{Aut}}(A))$, where $A$ is the polynomial or free associative algebra over the base field $K$. We prove that all Ind-scheme automorphisms of $\mathrm{\text{Aut}}(K[x_1,\dots ,x_n])$ are inner for $n\ge 3$, and all Ind-scheme automorphisms of $\mathrm{\text{Aut}}(K〈x_1,\dots ,x_n〉)$ are semi-inner. As an application, we prove that $\mathrm{\text{Aut}}(K[x_1,\dots ,x_n])$ cannot be embedded into $\mathrm{\text{Aut}}(K〈x_1,\dots ,x_n〉)$ by the natural abelianization. In other words, the Automorphism Group Lifting Problem has a negative solution. We explore close connection between the above results and the Jacobian conjecture, as well as the Kanel-Belov–Kontsevich conjecture, and formulate the Jacobian conjecture for fields of any characteristic. We make use of results of Bodnarchuk and Rips, and we also consider automorphisms of tame groups preserving the origin and obtain a modification of said results in the tame setting.

Let $L$ be a relatively free Lie algebra of finite rank $n$, with $n\ge 2$, $\widehat{L}$ be the completion of $L$ with respect to the topology defined by the lower central series $L={\gamma }_1(L)\supseteq {\gamma }_2(L)\supseteq \cdots$ of $L$ and $L_{(c)}=L/{\gamma }_c(L)$, with $c\ge 2$. We prove that, with respect to the formal power series topology, the automorphism group $\mathrm{\text{Aut}}(L)$ of $L$ is dense in the automorphism group $\mathrm{\text{Aut}}(\widehat{L})$ of $\widehat{L}$ if and only if $L$ is nilpotent. Furthermore, we show that there exists a dense subgroup of $\mathrm{\text{Aut}}(\widehat{L})$ generated by ${\mathrm{\text{GL}}}_n(K)$ and a finite set of IA-automorphisms if and only if $\mathrm{\text{Aut}}(L_{(c)})$ is generated by ${\mathrm{\text{GL}}}_n(K)$ and a finite set of IA-automorphisms independent upon $c$ for all $c$. We apply our study to several varieties of Lie algebras.

The additive primitive length of an element $f$ of a relatively free algebra $F_d(𝔙)$ in a variety of algebras $𝔙$ is equal to the minimal number $\ell$ such that $f$ can be presented as a sum of $\ell$ primitive elements. We give an upper bound for the additive primitive length of the elements in the $d$-generated polynomial algebra over a field of characteristic 0, $d>1$. The bound depends on $d$ and on the degree of the element. We show that if the field has more than two elements, then the additive primitive length in free $d$-generated nilpotent-by-abelian Lie algebras is bounded by 5 for $d=3$ and by 6 for $d>3$. If the field has two elements only, then our bounds are 6 for $d=3$ and 7 for $d>3$. This generalizes a recent result of Ela Aydın for two-generated free metabelian Lie algebras. In all cases considered in the paper, the presentation of the elements as sums of primitive elements can be found effectively in polynomial time.

For a positive integer $n\ge 4$, 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 $\mathrm{\text{Aut}}(R_n)$ generated by the tame automorphisms and a countably infinite set of explicitly given automorphisms of $R_n$ is dense in $\mathrm{\text{Aut}}(R_n)$ with respect to the formal power series topology.