This paper is the first in a sequence on the first order theory of free products and further generalizations. In the first paper, we generalize the analysis of systems of equations over free and (torsion-free) hyperbolic groups, and analyze systems of equations over free products. To do that we introduce limit groups over the class of free products, and show that a finitely presented group has a canonical (finite) collection of maximal limit quotients. We further extend this finite collection and associate a Makanin–Razborov diagram over free products with every f.p. group. This MR diagram encodes all the quotients of a given f.p. group that are free products, all its homomorphisms into free products, and equivalently all the solutions to a given system of equations over a free product.

In this article, we prove new results about the existence of $2$-cells in disc diagrams which are extreme in the sense that they are attached to the rest of the diagram along a small connected portion of their boundary cycle. In particular, we establish conditions on a $2$-complex $X$ which imply that all minimal area disc diagrams over $X$ with reduced boundary cycles have extreme $2$-cells in this sense. The existence of extreme $2$-cells in disc diagrams over these complexes leads to new results on coherence using the perimeter-reduction techniques we developed in an earlier article. Recall that a group is called coherent if all of its finitely generated subgroups are finitely presented. We illustrate this approach by showing that several classes of one-relator groups, small cancellation groups and groups with staggered presentations are collections of coherent groups.

Motivated by the work of McCarthy and Papadopoulos for subgroups of mapping class groups, we construct domains of proper discontinuity in the compactified Outer space and in the projectivized space of geodesic currents for any “dynamically large” subgroup of $\operatorname{Out}(F_N)$ (that is, a subgroup containing an atoroidal iwip).

As a corollary, we prove that for $N\ge3$ the action of $\operatorname{Out}(F_N)$ on the subset of $\mathbb{P}\operatorname{Curr}(F_N)$ consisting of all projectivized currents with full support is properly discontinuous.

We present an algorithm to convert a word of length $n$ in the standard generators of the solvable Baumslag–Solitar group $\operatorname{BS}(1,p)$ into a geodesic word, which runs in linear time and $O(n\log n)$ space on a random access machine.

We show that there is no algorithm deciding whether the maximal residually free quotient of a given finitely presented group is finitely presentable or not.

Given a finitely generated subgroup $G$ of a finite product of limit groups, we discuss the possibility of finding an explicit set of defining equations (i.e., of expressing $G$ as the maximal residually free quotient of an explicit finitely presented group).

Group presentations are implicit descriptions of $2$-dimensional cell complexes with only one vertex. While such complexes are usually sufficient for topological investigations of groups, multi-vertex complexes are often preferable when the focus shifts to geometric considerations. In this article, I show how to quickly describe the most important multi-vertex $2$-complexes using a slight variation of the traditional group presentation. As an illustration, I describe multi-vertex $2$-complexes for torus knot groups and one-relator Artin groups from which their elementary properties are easily derived. The latter are used to give an easy geometric proof of a classic result of Appel and Schupp.

We prove that any local irreducible automaton is contained in a complete one. The proof uses Nasu’s masking lemma. We use the same technique to give a new proof that any weakly deterministic irreducible automaton is contained in a complete one.

Let $F$ be a finitely generated free group. We present an algorithm such that, given a subgroup $H\leq F$, decides whether $H$ is the fixed subgroup of some family of automorphisms, or family of endomorphisms of $F$ and, in the affirmative case, finds such a family. The algorithm combines both combinatorial and geometric methods.

Sublinear time algorithms represent a new paradigm in computing, where an algorithm must give some sort of an answer after inspecting only a small portion of the input. The most typical situation where sublinear time algorithms are considered is property testing. There are several interesting contexts where one can test properties in sublinear time. A canonical example is graph colorability. To tell that a given graph is not $k$-colorable, it is often sufficient to inspect just one vertex with incident edges: if the degree of a vertex is greater than $k$, then the graph is not $k$-colorable.

It is a challenging and interesting task to find algebraic properties that could be tested in sublinear time. In this paper, we address several algorithmic problems in the theory of groups and semigroups that may admit sublinear time solution, at least for “most” inputs.

Radicals for Fitting pseudovarieties of groups are investigated from a profinite viewpoint in order to describe Mal’cev products on the left by the corresponding local pseudovariety of semigroups.

Let $(G_\ell\mid\ell\in L)$ be a family of groups and let $F$ be a free group. Let $G$ denote $F \ast {\Large{*}}_{\ell\in L} G_\ell$, the free product of $F$ and all the $G_\ell$. Let $\mathcal{F}$ denote the set of all finitely generated (free) subgroups $H$ of $G$ which have the property that, for each $g \in G$ and each $\ell\in L$, $H \cap G_\ell^{g} = \{1\}$. For each free group $H$, the reduced rank of $H$ is defined as $\bar{r}(H) := \max\{\operatorname{rank}(H) -1, 0\} \in[0,\infty]$. Set \begin{eqnarray*} \theta & := & \max\biggl\{ \biggl\vert\frac{\lvert D \rvert}{\lvert D \rvert-2}\biggr\vert : \\ &&{}\mbox{$D$ is a finite subgroup of $G$ with }\lvert D \rvert \ne2 \biggr\} \in [1,3],\\ \sigma & := & \inf\biggl\{ s \in[0,\infty] : \mbox{for all } H, K \in\mathcal{F}, \\ &&{}\sum_{HgK \in H \setminus G /K}\bar{r} (H^g\cap K) \le s {\theta} \bar{r} (H) \bar{r} (K) \biggr\} \in [0,\infty]. \end{eqnarray*} We are interested in precise bounds for $\sigma$. If every element of $\mathcal{F}$ is cyclic, then $\sigma = 0$. We henceforth assume that some element of $\mathcal{F}$ has rank two.

In the case where $G=F$ and, hence, $v = 1$, Hanna Neumann and Walter Neumann proved that $\sigma\in[1,2]$ and it is a famous conjecture that $\sigma = 1$, called the Strengthened Hanna Neumann Conjecture.

For the general case, we proved that $\sigma \in[1,2]$ and if $G$ has 2-torsion then $\sigma = 2$. We conjectured that if $G$ is 2-torsion-free then $\sigma = 1$. In this article, we prove the following implications which show that under certain circumstances $\sigma \lt 2$.

If $G$ is $2$-torsion-free and has $3$-torsion, then $\sigma \le \frac{8}{7}$.

If $G$ is $2$-torsion-free and $3$-torsion-free and has $5$-torsion, then $\sigma \le \frac{9}{5}$.

If $p$ is an odd prime number and $G = C_p \ast C_p$, then $\sigma \le 2 - \frac{(4+2\sqrt{3})p}{(2 p -3 + \sqrt{3})^2}$. In particular, if $G = C_3 \ast C_3$ then $\sigma = 1$, and if $G = C_5 \ast C_5$ then $\sigma \le 1.52$.

We study the automorphisms of graph products of cyclic groups, a class of groups that includes all right-angled Coxeter and right-angled Artin groups. We show that the group of automorphisms generated by partial conjugations is itself a graph product of cyclic groups providing its defining graph does not contain any separating intersection of links (SIL). In the case that all the cyclic groups are finite, this implies that the automorphism group is virtually $\operatorname{CAT}(0)$; it has a finite index subgroup which acts geometrically on a right-angled building.

We study properties of (bi-infinite) arrays having all adjacent $k\times k$ adjacent minors equal to one. If we further add the condition that all adjacent $(k-1)\times(k-1)$ minors be nonzero, then these arrays are necessarily of rank $k$. It follows that we can explicit construct all of them. Several nice properties are made apparent. In particular, we revisit, with this perspective, the notion of frieze patterns of Coxeter. This shed new light on their properties. A connexion is also established with the notion of $T$-systems of Statistical Physics.

To every finitely generated group, one can assign the conjugacy growth function that counts the number of conjugacy classes intersecting a ball of radius $n$. Results of Ivanov and Osin show that the conjugacy growth function may be constant even if the (ordinary) growth function is exponential. The aim of this paper is to provide conjectures, examples and statements that show that in “normal” cases, groups with exponential growth functions also have exponential conjugacy growth functions.

The objective of this note is to give some alternative proofs that the intersection of the normal subgroups with torsion-free nilpotent factor groups of free groups is the identity. Our approach gives rise to a new family of one-relator groups with this property. This family includes, among others, the fundamental groups of orientable surfaces. Our hope is that the idea involved will lead to new insights into these well-known groups, and, even more important for us, a proof that free $\mathbb{Q}$-groups have the same property.

We start by studying the distribution of (cyclically reduced) elements of the free groups $F_n$ with respect to their Abelianization (or equivalently, their class in $H_1(F_n, \mathbf{Z})$). We derive an explicit generating function, and a limiting distribution, by means of certain results (of independent interest) on Chebyshev polynomials; we also prove that the reductions mod $\mod p$ ($p$—an arbitrary prime) of these classes are asymptotically equidistributed, and we study the deviation from equidistribution. We extend our techniques to a more general setting and use them to study the statistical properties of long cycles (and paths) on regular (directed and undirected) graphs. We return to the free group to study some growth functions of the number of conjugacy classes as a function of their cyclically reduced length.

In this paper, we study the generic, i.e., typical, behavior of finitely generated subgroups of hyperbolic groups and also the generic behavior of the word problem for amenable groups. We show that a random set of elements of a nonelementary word hyperbolic group is very likely to be a set of free generators for a nicely embedded free subgroup. We also exhibit some finitely presented amenable groups for which the restriction of the word problem is unsolvable on every sufficiently large subset of words.

We prove that a finitely generated solvable group which is not virtually nilpotent has exponential conjugacy growth.