On a question of Bumagin and Wise

Motivated by a question of Bumagin and Wise, we construct a continuum of finitely generated, residually finite groups whose outer automorphism groups are pairwise non-isomorphic finitely generated, non-recursively-presentable groups. These are the first examples of such residually finite groups.Comment: 8 page

Every group is the outer automorphism group of an HNN-extension of a fixed triangle group

Fix an equilateral triangle group $T_i=\langle a, b; a^i, b^i, (ab)^i\rangle$ with $i\geq6$ arbitrary. Our main result is: for every presentation $\mathcal{P}$ of every countable group $Q$ there exists an HNN-extension $T_{\mathcal{P}}$ of $T_i$ such that $\operatorname{Out}(T_{\mathcal{P}})\cong Q$. We construct the HNN-extensions explicitly, and examples are given. The class of groups constructed have nice categorical and residual properties. In order to prove our main result we give a method for recognising malnormal subgroups of small cancellation groups, and we introduce the concept of "malcharacteristic" subgroups.Comment: 39 pages. Final version, to appear in Advances in Mathematic

On the outer automorphism groups of finitely generated, residually finite groups

Bumagin-Wise posed the question of whether every countable group can be realised as the outer automorphism group of a finitely generated, residually finite group. We give a partial answer to this problem for recursively presentable groups.Comment: 13 pages. Final versio

The Outer Automorphism Groups of Two-Generator One-Relator Groups with Torsion

The main result of this paper is a complete classification of the outer automorphism groups of two-generator, one-relator groups with torsion. To this classification we apply recent algorithmic results of Dahmani--Guirardel, which yields an algorithm to compute the isomorphism class of the outer automorphism group of a given two-generator, one-relator group with torsion.Comment: 15 pages, final version. To appear in Proc. Amer. Math. So

The Equalizer Conjecture for the free group of rank two

Funding: This research was supported by Engineering and Physical Sciences Research Council (EPSRC) grant EP/R035814/1.The equalizer of a set of homomorphisms S : F(a,b) → F(Δ) has rank at most two if S contains an injective map and is not finitely generated otherwise. This proves a strong form of Stallings’ Equalizer Conjecture for the free group of rank two. Results are also obtained for pairs of homomorphisms g,h : F(Σ) → F(Δ) when the images are inert in, or retracts of, F(Δ).Publisher PDFPeer reviewe

The residual finiteness of (hyperbolic) automorphism-induced HNN-extensions

We classify finitely generated, residually finite automorphism-induced HNN-extensions in terms of the residual separability of a single associated subgroup. This classification provides a method to construct automorphism-induced HNN-extensions which are not residually finite. We prove that this method can never yield a “new” counter-example to Gromov’s conjecture on the residual finiteness of hyperbolic groups

The Conjugacy Problem for ascending HNN-extensions of free groups

We give an algorithm to solve the Conjugacy Problem for ascending HNN-extensions of free groups. To do this, we give algorithms to solve certain problems on dynamics of free group endomorphisms.Comment: 30 pages, v2: rearranging sections to improve expositio

The JSJ-decompositions of one-relator groups with torsion

In this paper we use JSJ-decompositions to formalise a folk conjecture recorded by Pride on the structure of one-relator groups with torsion. We prove a slightly weaker version of the conjecture, which implies that the structure of one-relator groups with torsion closely resemble the structure of torsion-free hyperbolic groups.Comment: 21 pages, 2 figures. Final versio

Holocene reefs and sediments of Castle Harbour, Bermuda

Quantitative studies on the corals of Holocene fringing, pinnacle and knoll coral-algal reefs of Castle Harbour, Bermuda, show that only about 10% of the available reef substrate is coralcovered. Of this, about 70% of the total coral-covered area is occupied by corals belonging to the Madracis-Oculina assemblage...