Let $H\le F$ be two finitely generated free groups. Given $g\in F$, we study the ideal $\mathfrak I_g$ of equations for g with coefficients in H, i.e. the elements $w(x)\in H*\langle x\rangle$ such that $w(g)=1$ in F. The ideal $\mathfrak I_g$ is a normal subgroup of $H*\langle x\rangle$, and it’s possible to algorithmically compute a finite normal generating set for $\mathfrak I_g$; we give a description of one such algorithm, based on Stallings folding operations. We provide an algorithm to find an equation in w(x)\in$\mathfrak I_g$ with minimum degree, i.e.  such that its cyclic reduction contains the minimum possible number of occurrences of x and x−1; this answers a question of A. Rosenmann and E. Ventura. More generally, we show how to algorithmically compute the set Dg of all integers d such that $\mathfrak I_g$ contains equations of degree d; we show that Dg coincides, up to a finite set, with either $\mathbb N$ or $2\mathbb N$. Finally, we provide examples to illustrate the techniques introduced in this paper. We discuss the case where ${\text{rank}}(H)=1$. We prove that both kinds of sets Dg can actually occur. We show that the equations of minimum possible degree aren’t in general enough to generate the whole ideal $\mathfrak I_g$ as a normal subgroup.

We introduce two families of two-generator one-relator groups called primitive extension groups and show that a one-relator group is hyperbolic if its primitive extension subgroups are hyperbolic. This reduces the problem of characterizing hyperbolic one-relator groups to characterizing hyperbolic primitive extension groups. These new groups, moreover, admit explicit decompositions as graphs of free groups with adjoined roots. In order to obtain this result, we characterize $2$-free one-relator groups with exceptional intersection in terms of Christoffel words, show that hyperbolic one-relator groups have quasi-convex Magnus subgroup, and build upon the one-relator tower machinery developed in previous work of the author.

Given a finitely generated free group $ {\mathbb {F} }$ of $\mathsf {rank}( {\mathbb {F} } )\geq 3$, we show that the mapping torus of $\phi$ is (strongly) relatively hyperbolic if $\phi$ is exponentially growing. As a corollary of our work, we give a new proof of Brinkmann's theorem which proves that the mapping torus of an atoroidal outer automorphism is hyperbolic. We also give a new proof of the Bridson–Groves theorem that the mapping torus of a free group automorphism satisfies the quadratic isoperimetric inequality. Our work also solves a problem posed by Minasyan and Osin: the mapping torus of an outer automorphism is not virtually acylindrically hyperbolic if and only if $\phi$ has finite order.

For a group $G$, let $\unicode[STIX]{x1D6E4}(G)$ denote the graph defined on the elements of $G$ in such a way that two distinct vertices are connected by an edge if and only if they generate $G$. Let $\unicode[STIX]{x1D6E4}^{\ast }(G)$ be the subgraph of $\unicode[STIX]{x1D6E4}(G)$ that is induced by all the vertices of $\unicode[STIX]{x1D6E4}(G)$ that are not isolated. We prove that if $G$ is a 2-generated noncyclic abelian group, then $\unicode[STIX]{x1D6E4}^{\ast }(G)$ is connected. Moreover, $\text{diam}(\unicode[STIX]{x1D6E4}^{\ast }(G))=2$ if the torsion subgroup of $G$ is nontrivial and $\text{diam}(\unicode[STIX]{x1D6E4}^{\ast }(G))=\infty$ otherwise. If $F$ is the free group of rank 2, then $\unicode[STIX]{x1D6E4}^{\ast }(F)$ is connected and we deduce from $\text{diam}(\unicode[STIX]{x1D6E4}^{\ast }(\mathbb{Z}\times \mathbb{Z}))=\infty$ that $\text{diam}(\unicode[STIX]{x1D6E4}^{\ast }(F))=\infty$.

We prove Wise's $W$-cycles conjecture. Consider a compact graph $\Gamma '$ immersing into another graph $\Gamma $. For any immersed cycle $\Lambda :{{S}^{1}}\to \Gamma $, we consider the map $\Lambda '$ from the circular components $\mathbb{S}$ of the pullback to $\Gamma '$. Unless $\Lambda '$ is reducible, the degree of the covering map $\mathbb{S}\to {{S}^{1}}$ is bounded above by minus the Euler characteristic of $\Gamma '$. As a corollary, any finitely generated subgroup of a one-relator group has a finitely generated Schur multiplier.

Let G be a group of odd order that contains a non-central element x whose order is either a prime p ≥ 5 or 3l, with l ≥ 2. Then, in , the group of units of ℤG, we can find an alternating unit u based on x, and another unit v, which can be either a bicyclic or an alternating unit, such that for all sufficiently large integers m we have that 〈um, vm〉 = 〈um〉 ∗ 〈vm〉 ≌ ℤ ∗ ℤ

We show that the restriction of the Dehornoy ordering to an appropriate free subgroup of the three-strand braid group defines a left-ordering of the free group on k generators, k>1, that has no convex subgroups.

This note contains some remarks on generating pairs for automorphism groups of free groups. There has been significant use of electronic assistance. Little of this is used to verify the results.

We revisit the problem of deciding whether a finitely generated subgroup H is a free factor of a given free group F. Known algorithms solve this problem in time polynomial in the sum of the lengths of the generators of H and exponential in the rank of F. We show that the latter dependency can be made exponential in the rank difference rank(F) - rank(H), which often makes a significant change.

Let G be a group, let RG be the group ring of the group G over the unital commutative ring R and let U(RG) be its group of units. Conditions which imply that U(RG) contains no free noncyclic group are studied, when R is a field of characteristic p ≠ 0, not algebraic over its prime field, and G is a solvable-by-finite group without p-elements. We also consider the case R = ℤp, the ring of p-adic integers and G torsionby- nilpotent torsion free group. Finally, the residual nilpotence of U(ℤpG) is investigated.