We examine a cyclic order on the directed edges of a tree whose vertices have cyclically ordered links. We use it to show that a graph of groups with left-cyclically ordered vertex groups and convex left-ordered edge groups is left-cyclically orderable.

We study a family of finitely generated residually finite small-cancellation groups. These groups are quotients of $F_2$ depending on a subset $S$ of positive integers. Varying $S$ yields continuously many groups up to quasi-isometry.

We examine 2-complexes $X$ with the property that for any compact connected $Y$, and immersion $Y\rightarrow X$, either $\unicode[STIX]{x1D712}(Y)\leqslant 0$ or $\unicode[STIX]{x1D70B}_{1}Y=1$. The mapping torus of an endomorphism of a free group has this property. Every irreducible 3-manifold with boundary has a spine with this property. We show that the fundamental group of any 2-complex with this property is locally indicable. We outline evidence supporting the conjecture that this property implies coherence. We connect the property to asphericity. Finally, we prove coherence for 2-complexes with a stricter form of this property. As a corollary, every one-relator group with torsion is coherent.

A tubular group G is a finite graph of groups with ℤ2 vertex groups and ℤ edge groups. We characterize residually finite tubular groups: G is residually finite if and only if its edge groups are separable. Methods are provided to determine if G is residually finite. When G has a single vertex group an algorithm is given to determine residual finiteness.

We show that certain right-angled Coxeter groups have finite index subgroups that quotient to $\mathbb{Z}$ with finitely generated kernels. The proof uses Bestvina–Brady Morse theory facilitated by combinatorial arguments. We describe a variety of examples where the plan succeeds or fails. Among the successful examples are the right-angled reflection groups in $\mathbb{H}^{4}$ with fundamental domain the 120-cell or the 24-cell.

We prove that non-elementary hyperbolic groups grow exponentially more quickly than their infinite index quasiconvex subgroups. The proof uses the classical tools of automatic structures and Perron–Frobenius theory.

We also extend the main result to relatively hyperbolic groups and cubulated groups. These extensions use the notion of growth tightness and the work of Dahmani, Guirardel and Osin on rotating families.

We show that certain graphs of groups with cyclic edge groups are aTmenable. In particular, this holds when each vertex group is either virtually special or acts properly and semisimply on ℍn.

We prove that if $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}S$ is a properly embedded $\pi _1$-injective surface in a compact 3-manifold $M$, then $\pi _1S$ is separable in $\pi _1M$.

We give a generalized and self-contained account of Haglund–Paulin’s wallspaces and Sageev’s construction of the CAT(0) cube complex dual to a wallspace. We examine criteria on a wallspace leading to finiteness properties of its dual cube complex. Our discussion is aimed at readers wishing to apply these methods to produce actions of groups on cube complexes and understand their nature. We develop the wallspace ideas in a level of generality that facilitates their application. Our main result describes the structure of dual cube complexes arising from relatively hyperbolic groups. Let $H_1,\ldots, H_s$ be relatively quasiconvex codimension-1 subgroups of a group $G$ that is hyperbolic relative to $P_1, \ldots, P_r$. We prove that $G$ acts relatively cocompactly on the associated dual CAT(0) cube complex $C$. This generalizes Sageev’s result that $C$ is cocompact when $G$ is hyperbolic. When $P_1,\ldots, P_r$ are abelian, we show that the dual CAT(0) cube complex $C$ has a $G$-cocompact CAT(0) truncation.

Let 〈a1, . . ., am ∣ wn〉 be a presentation of a group G, where n ≥ 2. We define a system of codimension-1 subspaces in the universal cover, and invoke Sageev's construction to produce an action of G on a CAT(0) cube complex. We show that the action is proper and cocompact when n ≥ 4. A fundamental tool is a geometric generalization of Pride's C(2n) small-cancellation result. We prove similar results for staggered groups with torsion.

a tits alternative theorem is proved in this paper for groups acting on cat(0) cubical complexes. that is, a proof is given to show that if $g$ is assumed to be a group for which there is a bound on the orders of its finite subgroups, and if $g$ acts properly on a finite-dimensional cat(0) cubical complex, then either $g$ contains a free subgroup of rank 2, or $g$ is finitely generated and virtually abelian. in particular, the above conclusion holds for any group $g$ with a free action on a finite-dimensional cat(0) cubical complex.

a conjecture is proposed, bounding the number of cycles with label $w^n$ in a labeled directed graph. some partial results towards this conjecture are established. as a consequence, it is proved that $\langle a_1, a_2, \ldots\,{\mid}\,w^n\rangle$ is coherent for $n\,{\geq}\,4$. furthermore, it is coherent for $n\,{\geq}\,2$, provided that the strengthened hanna neumann conjecture holds.

We investigate the problem of whether every immersed flat plane in a nonpositively curved square complex is the limit of periodic flat planes. Using a branched cover, we reduce the problem to the case of $\mathcal{V}\mathcal{H}$-complexes. We solve the problem for malnormal and cyclonormal $\mathcal{V}\mathcal{H}$-complexes. We also solve the problem for complete square complexes using a different approach. We give an application towards deciding whether the elements of fundamental groups of the spaces we study have commuting powers. We note a connection between the flat approximation problem and subgroup separability.

For each finitely presented group $Q$ , a short exact sequence $1 \rightarrow N \rightarrow G \rightarrow Q \rightarrow 1$ is produced, such that $G$ is residually finite and $N$ is finitely generated.

A criterion is given for showing that certain one-relator groups are residually finite. This is applied to a one-relator group with torsion $G = \langle a_1, \ldots, a_r \mid W^n\rangle$ . It is shown that $G$ is residually finite provided that $W$ is outside the commutator subgroup and $n$ is sufficiently large. An important ingredient in the proof is a criterion which implies that a subgroup of a group is malnormal. A graded small-cancellation criterion is developed which detects whether a map $A \rightarrow B$ between graphs induces a $\pi_1$ -injection, and whether $\pi_1 A$ maps to a malnormal subgroup of $\pi_1 B$ .

This paper provides a strengthening of the theorems of small cancellation theory. It is proven that disc diagrams contain 'fans' of consecutive 2-cells along their boundaries. The size of these fans is linked to the strength of the small cancellation conditions satisfied by the diagram. A classification result is proven for disc diagrams satisfying small cancellation conditions. Any disc diagram either contains three fans along its boundary, or it is a ladder, or it is a wheel. Similar statements are proven for annular diagrams.

2000 Mathematical Subject Classification: 20F06.

The Spelling Theorem of B. B. Newman states that for a one-relator group (a1, … | Wn), any nontrivial word which represents the identity must contain a (cyclic) subword of W±n longer than Wn−1. We provide a new proof of the Spelling Theorem using towers of 2-complexes. We also give a geometric classification of reduced disc diagrams in one-relator groups with torsion. Either the disc diagram has three 2-cells which lie almost entuirly along the bounday, or the disc diagram looks like a ladder. We use this ladder theorem to prove that a large class of one-relator groups with torsion are locally quasiconvex.

It is shown that every ascending HNN extension of a finitely generated free group is Hopfian. An important ingredient in the proof is that under certain hypotheses on the group H, if G is an ascending HNN extension of H, then cd(G) = cd(H) + 1.