Bordered Heegaard Floer homology is a three-manifold invariant which associates to a surface $F$ an algebra $\mathcal{A}\left(F\right)$ and to a three-manifold $Y$ with boundary identified with $F$ a module over $\mathcal{A}\left(F\right)$. In this paper, we establish naturality properties of this invariant. Changing the diffeomorphism between $F$ and the boundary of $Y$ tensors the bordered invariant with a suitable bimodule over $\mathcal{A}\left(F\right)$. These bimodules give an action of a suitably based mapping class group on the category of modules over $\mathcal{A}\left(F\right)$. The Hochschild homology of such a bimodule is identified with the knot Floer homology of the associated open book decomposition. In the course of establishing these results, we also calculate the homology of $\mathcal{A}\left(F\right)$. We also prove a duality theorem relating the two versions of the $3$–manifold invariant. Finally, in the case of a genus-one surface, we calculate the mapping class group action explicitly. This completes the description of bordered Heegaard Floer homology for knot complements in terms of the knot Floer homology.

The study of limit groups, that is, finitely generated fully residually free groups, was a key first step towards the understanding of the elementary theory of a free group. In this paper we conduct a systematic study of the class $\mathfrak{U}$ of finitely generated fully residually partially commutative groups.

Our first main goal is to give an algebraic characterisation of the class $\mathfrak{U}$: a finitely generated group $G$ is fully residually partially commutative if and only if it is a subgroup of a graph tower (a group built hierarchically using partially commutative groups and (nonexceptional) surfaces.) Furthermore, if the group $G$ is given by its finite radical presentation, then the graph tower and the embedding can be effectively constructed. This result generalises the work of Kharlampovich and Miasnikov on fully residually free groups.

Following Sela’s approach to limit groups, the second goal of the paper is to provide a dynamical characterisation of the class $\mathfrak{U}$. We introduce a class of spaces, called real cubings, as higher-dimensional generalisations of real trees and show that a specific type of action on these spaces characterises the class $\mathfrak{U}$: a finitely generated group acts freely cospecially on a real cubing if and only if it is fully residually partially commutative. As a corollary we get that (geometric) limit groups over partially commutative groups are fully residually partially commutative. This result generalises the work of Sela on limit groups over free groups.

It is conjectured that, in the asymptotic expansion of the Kashaev invariant of a hyperbolic knot, the first coefficient is represented by the complex volume of the knot complement, and the second coefficient is represented by a constant multiple of the square root of the twisted Reidemeister torsion associated with the holonomy representation of the hyperbolic structure of the knot complement. In particular, this conjecture has been rigorously proved for some simple hyperbolic knots, for which the second coefficient is presented by a modification of the square root of the Hessian of the potential function of the hyperbolic structure of the knot complement.

In this paper, we define an invariant of a parametrized knot diagram as a modification of the Hessian of the potential function obtained from the parametrized knot diagram. Further, we show that this invariant is equal (up to sign) to a constant multiple of the twisted Reidemeister torsion for any two-bridge knot.

We give sharp upper bounds on the maximal injectivity radius of finite-area hyperbolic surfaces and use them, for each $g\ge 2$, to identify a constant $r_{g-1,2}$ such that the set of closed genus-$g$ hyperbolic surfaces with maximal injectivity radius at least $r$ is compact if and only if $r>r_{g-1,2}$. The main tool is a version of the centered dual complex that we introduced earlier, a coarsening of the Delaunay complex. In particular, we bound the area of a compact centered dual two-cell below given lower bounds on its side lengths.

We prove a version of the classical Runge and Mergelyan uniform approximation theorems for nonorientable minimal surfaces in Euclidean $3$–space ${ℝ}^3$. Then we obtain some geometric applications. Among them, we emphasize the following ones:

• A Gunning–Narasimhan-type theorem for nonorientable conformal surfaces.

• An existence theorem for nonorientable minimal surfaces in ${ℝ}^3$ with arbitrary conformal structure, properly projecting into a plane.

• An existence result for nonorientable minimal surfaces in ${ℝ}^3$ with arbitrary conformal structure and Gauss map omitting one projective direction.

Let ${\mathcal{C}}_{TS}$ be the subgroup of the smooth knot concordance group generated by topologically slice knots. Endo showed that ${\mathcal{C}}_{TS}$ contains an infinite-rank subgroup, and Livingston and Manolescu-Owens showed that ${\mathcal{C}}_{TS}$ contains a ${\mathbb{Z}}^3$ summand. We show that in fact ${\mathcal{C}}_{TS}$ contains a ${\mathbb{Z}}^{\infty }$ summand. The proof relies on the knot Floer homology package of Ozsváth–Szabó and the concordance invariant $\epsilon$.

Let $\varphi \in Out\left(F_n\right)$ be a free group outer automorphism that can be represented by an expanding, irreducible train-track map. The automorphism $\varphi$ determines a free-by-cyclic group $\Gamma =F_n{⋊}_{\varphi }\mathbb{Z}$ and a homomorphism $\alpha \in H^1\left(\Gamma ;\mathbb{Z}\right)$. By work of Neumann, Bieri, Neumann and Strebel, and Dowdall, Kapovich and Leininger, $\alpha$ has an open cone neighborhood $\mathcal{A}$ in $H^1\left(\Gamma ;ℝ\right)$ whose integral points correspond to other fibrations of $\Gamma$ whose associated outer automorphisms are themselves representable by expanding irreducible train-track maps. In this paper, we define an analog of McMullen’s Teichmüller polynomial that computes the dilatations of all outer automorphisms in $\mathcal{A}$.