The holomorphic couch theorem

We prove that if two conformal embeddings between Riemann surfaces with finite topology are homotopic, then they are isotopic through conformal embeddings. Furthermore, we show that the space of all conformal embeddings in a given homotopy class deformation retracts into a point, a circle, a torus, or the unit tangent bundle of the codomain, depending on the induced homomorphism on fundamental groups. Quadratic differentials play a central role in the proof.Comment: 70 pages, 13 figures. Sections 4 and 8 modified following referee's repor

Conformal grafting and convergence of Fenchel-Nielsen twist coordinates

We cut a hyperbolic surface of finite area along some analytic simple closed curves, and glue in cylinders of varying moduli. We prove that as the moduli of the glued cylinders go to infinity, the Fenchel-Nielsen twist coordinates for the resulting surface around those cylinders converge.Comment: 18 pages, 4 figure

The converse of the Schwarz Lemma is false

Let h : X → Y be a homeomorphism between hyperbolic surfaces with finite topology. If h is homotopic to a holomorphic map, then every closed geodesic in X is at least as long as the corresponding geodesic in Y, by the Schwarz Lemma. The converse holds trivially when X and Y are disks or annuli, and it holds when X and Y are closed surfaces by a theorem of Thurston. We prove that the converse is false in all other cases, strengthening a result of Masumoto

Failure of the well-rounded retract for Outer space and Teichm\"uller space

The well-rounded retract for $\mathrm{SL}_n(\mathbb{Z})$ is defined as the set of flat tori of unit volume and dimension $n$ whose systoles generate a finite-index subgroup in homology. This set forms an equivariant spine of minimal dimension for the space of flat tori. For both the Outer space $X_n$ of metric graphs of rank $n$ and the Teichm\"uller space $\mathcal{T}_g$ of closed hyperbolic surfaces of genus $g$, we show that the literal analogue of the well-rounded retract does not contain an equivariant spine. We also prove that the set of graphs whose systoles fill (the analogue of a set proposed as a spine for $\mathcal{T}_g$ by Thurston) is a spine for $X_n$ but that its dimension is larger than the virtual cohomological dimension of $\mathrm{Out}(F_n)$ in general.Comment: 8 page