Narrow Results

We first present a Skoda-type division theorem for holomorphic sections of line bundles on a projective variety which is essentially the most general, compared to previous ones. Then we revisit Geometric Effective Nullstellensatz and observe that even this general Skoda division is far from sufficient to yield stronger GEN such as ‘vanishing order $1$ division’, which could be used for finite generation of section rings by the basic finite generation lemma. To resolve this problem, we develop a notion of pseudo-division and show that it can replace the usual division in the finite generation lemma. We also give a vanishing order 1 pseudo-division result when the line bundle is ample.

A self-similar group of finite type is the profinite group of all automorphisms of a regular rooted tree that locally around every vertex act as elements of a given finite group of allowed actions. We provide criteria for determining when a self-similar group of finite type is finite, level-transitive, or topologically finitely generated. Using these criteria and GAP computations we show that for the binary alphabet there is no infinite topologically finitely generated self-similar group given by patterns of depth 3, and there are 32 such groups for depth 4.

We solve the subgroup intersection problem (SIP) for any RAAG $G$ of Droms type (i.e. with defining graph not containing induced squares or paths of length $3$): there is an algorithm which, given finite sets of generators for two subgroups $H,K\le G$, decides whether $H\cap K$ is finitely generated or not, and, in the affirmative case, it computes a set of generators for $H\cap K$. Taking advantage of the recursive characterization of Droms groups, the proof consists in separately showing that the solvability of SIP passes through free products, and through direct products with free-abelian groups. We note that most of RAAGs are not Howson, and many (e.g. ${𝔽}_2\times {𝔽}_2$) even have unsolvable SIP.

The singular braids with $n$ strands, $n\ge 3$, were introduced independently by Baez and Birman. It is known that the monoid formed by the singular braids is embedded in a group that is known as singular braid group, denoted by $S{G}_n$. There has been another generalization of braid groups, denoted by $GV{B}_n$, $n\ge 3$, which was introduced by Fang as a group of symmetries behind quantum quasi-shuffle structures. The group $GV{B}_n$ simultaneously generalizes the classical braid group, as well as the virtual braid group on $n$ strands.

We investigate the commutator subgroups $S{G}_n^{\prime }$ and $GV{B}_n^{\prime }$ of these generalized braid groups. We prove that $S{G}_n^{\prime }$ is finitely generated if and only if $n\ge 5$, and $GV{B}_n^{\prime }$ is finitely generated if and only if $n\ge 4$. Further, we show that both $S{G}_n^{\prime }$ and $GV{B}_n^{\prime }$ are perfect if and only if $n\ge 5$.

In [Finite presentability of Bruck–Reilly extensions of groups, J. Algebra242 (2001) 20–30], Araujo and Ruškuc studied finite generation and finite presentability of Bruck–Reilly extension of a group. In this paper, we aim to generalize some results given in that paper to generalized Bruck–Reilly ∗-extension of a group. In this way, we determine necessary and sufficent conditions for generalized Bruck–Reilly ∗-extension of a group, ${GBR}^{\ast }(G;\beta ,\gamma ;u)$, to be finitely generated and finitely presented. Let $G$ be a group, $\beta ,\gamma :G\to H_1^{\ast }$ be morphisms and $u\in H_1$ ($H_1^{\ast }$ and $H_1$ are the ${{\mathscr{H}}}^{\ast }$- and ${\mathscr{H}}$-classes, respectively, contains the identity element $1_T$ of $T$). We prove that ${GBR}^{\ast }(G;\beta ,\gamma ;u)$ is finitely generated if and only if there exists a finite subset $X_0\subseteq G$ such that $G$ is generated by $({\bigcup }_{i\ge 0}{X}_0{\beta }^i)\cup ({\bigcup }_{j\ge 0}{X}_0{\gamma }^j)$. We also prove that ${GBR}^{\ast }(G;\beta ,\gamma ;u)$ is finitely presented if and only if $G$ is presented by $〈X;R〉$, where $X$ is a finite set and