A fibering theorem for 3-manifolds

We generalize a result of Moon on the fibering of certain 3-manifolds over the circle. Our main theorem is the following: Let $M$ be a closed 3-manifold. Suppose that $G=\pi_1(M)$ contains a finitely generated group $U$ of infinite index in $G$ which contains a non-trivial subnormal subgroup $N\neq \mathbb{Z}$ of $G$, and suppose that $N$ has a composition series of length $n$ in which at least $n-1$ terms are finitely generated. Suppose that $N$ intersects nontrivially the fundamental groups of the splitting tori given by the Geometrization Theorem and that the intersections of $N$ with the fundamental groups of the geometric pieces are non-trivial and not isomorphic to $\mathbb{Z}$. Then, $M$ has a finite cover which is a bundle over $\mathbb{S}$ with fiber a compact surface $F$ such that $\pi_1(F)$ and $U$ are commensurable.

Numerical upper bounds on growth of automaton groups

The growth of a finitely generated group is an important geometric invariant which has been studied for decades. It can be either polynomial, for a well-understood class of groups, or exponential, for most groups studied by geometers, or intermediate, that is between polynomial and exponential. Despite recent spectacular progresses, the class of groups with intermediate growth remains largely mysterious. Many examples of such groups are constructed using Mealy automata. The aim of this paper is to give an algorithmic procedure to study the growth of such automaton groups, and more precisely to provide numerical upper bounds on their exponents. Our functions retrieve known optimal bounds on the famous first Grigorchuk group. They also improve known upper bounds on other automaton groups and permitted us to discover several new examples of automaton groups of intermediate growth. All the algorithms described are implemented in GAP, a language dedicated to computational group theory.

Group-theoretical graph categories

The semidirect product of a finitely generated group dual with the symmetric group can be described through so-called group-theoretical categories of partitions (covers only a special case; due to Raum–Weber, 2015) and skew categories of partitions (more general; due to Maaßen, 2018). We generalize these results to the case of graph categories, which allows to replace the symmetric group by the group of automorphisms of some graph.

BNS INVARIANTS AND ALGEBRAIC FIBRATIONS OF GROUP EXTENSIONS

Let G be a finitely generated group that can be written as an extension $$ \begin{align*} 1 \longrightarrow K \stackrel{i}{\longrightarrow} G \stackrel{f}{\longrightarrow} \Gamma \longrightarrow 1 \end{align*} $$ where K is a finitely generated group. By a study of the Bieri–Neumann–Strebel (BNS) invariants we prove that if $b_1(G)> b_1(\Gamma ) > 0$ , then G algebraically fibres; that is, admits an epimorphism to $\Bbb {Z}$ with finitely generated kernel. An interesting case of this occurrence is when G is the fundamental group of a surface bundle over a surface $F \hookrightarrow X \rightarrow B$ with Albanese dimension $a(X) = 2$ . As an application, we show that if X has virtual Albanese dimension $va(X) = 2$ and base and fibre have genus greater that $1$ , G is noncoherent. This answers for a broad class of bundles a question of J. Hillman ([9, Question 11(4)]). Finally, we show that there exist surface bundles over a surface whose BNS invariants have a structure that differs from that of Kodaira fibrations, determined by T. Delzant.

Groups with context-free Diophantine problem

We find algebraic conditions on a group equivalent to the position of its Diophantine problem in the Chomsky Hierarchy. In particular, we prove that a finitely generated group has a context-free Diophantine problem if and only if it is finite.

Groups with context-free Diophantine problem

We find algebraic conditions on a group equivalent to the position of its Diophantine problem in the Chomsky Hierarchy. In particular, we prove that a finitely generated group has a context-free Diophantine problem if and only if it is finite.

Profinite groups in which the probabilistic zeta function has no negative coefficients

To a finitely generated profinite group [Formula: see text], a formal Dirichlet series [Formula: see text] is associated, where [Formula: see text] and [Formula: see text] denotes the Möbius function of the lattice of open subgroups of [Formula: see text] Its formal inverse [Formula: see text] is the probabilistic zeta function of [Formula: see text]. When [Formula: see text] is prosoluble, every coefficient of [Formula: see text] is nonnegative. In this paper we discuss the general case and we produce a non-prosoluble finitely generated group with the same property.

Operator algebras with hyperarithmetic theory

We show that the following operator algebras have hyperarithmetic theory: the hyperfinite II$_1$ factor $\mathcal R$, $L(\varGamma )$ for $\varGamma $ a finitely generated group with solvable word problem, $C^*(\varGamma )$ for $\varGamma $ a finitely presented group, $C^*_\lambda (\varGamma )$ for $\varGamma $ a finitely generated group with solvable word problem, $C(2^\omega )$ and $C(\mathbb P)$ (where $\mathbb P$ is the pseudoarc). We also show that the Cuntz algebra $\mathcal O_2$ has a hyperarithmetic theory provided that the Kirchberg embedding problems have affirmative answers. Finally, we prove that if there is an existentially closed (e.c.) II$_1$ factor (resp. $\textrm{C}^*$-algebra) that does not have hyperarithmetic theory, then there are continuum many theories of e.c. II$_1$ factors (resp. e.c. $\textrm{C}^*$-algebras).

The Klein bottle group is not strongly verbally closed, though awfully close to being so

According to Mazhuga’s theorem, the fundamental group H of anyconnected surface, possibly except for the Klein bottle, is a retract of each finitely generated group containing H as a verbally closed subgroup. We prove that the Klein bottle group is indeed an exception but has a very close property.

On knot groups acting on trees

A finitely generated group [Formula: see text] acting on a tree with infinite cyclic edge and vertex stabilizers is called a generalized Baumslag–Solitar group (GBS group). We prove that a one-knot group [Formula: see text] is a GBS group if and only if [Formula: see text] is a torus knot group, and describe all n-knot GBS groups for [Formula: see text].