Christopher J. Leininger

In this paper we construct examples of Weil-Petersson geodesics with nonminimal ending laminations which have 1-dimensional limit sets in the Thurston compactification of Teichm\"{u}ller space.

In this paper we provide a negative answer to a question of Farb about the relation between the algebraic degree of the stretch factor of a pseudo-Anosov homeomorphism and the genus of the surface on which it is defined.

In this paper we prove that the space of flat metrics (nonpositively curved Euclidean cone metrics) on a closed, oriented surface is marked-length-spectrally rigid. In other words, two flat metrics assigning the same lengths to all closed curves differ by an isometry isotopic to the identity. The novel proof suggests a stronger rigidity conjecture for this class of metrics.

Any endomorphism of a finitely generated free group naturally descends to an injective endomorphism of its stable quotient. In this paper, we prove a geometric incarnation of this phenomenon: namely, that every expanding irreducible train track map inducing an endomorphism of the fundamental group gives rise to an expanding irreducible train track representative of the injective endomorphism of the stable quotient. As an application, we prove that the property of having fully irreducible monodromy for a splitting of a hyperbolic free-by-cyclic group depends only on the component of the BNS-invariant containing the associated homomorphism to the integers.

For d < 5, we describe an elementary construction of nonzero degree, strict contractions between closed, oriented hyperbolic d-oribifolds. Appealing to work of Gueritaud-Kassel and Tholozan, these examples determine exotic quotients of SO_0(d,1).

Consider a group G and an epimorphism u_0:G\to\Z inducing a splitting of G as a semidirect product ker(u_0)\rtimes_\varphi\Z with ker(u_0) a finitely generated free group and \varphi\in Out(ker(u_0)) representable by an expanding irreducible train track map. Building on our earlier work [Dynamics on free-by-cyclic groups, arXiv:1301.7739], in which we we realized G as \pi_1(X) for an Eilenberg-Maclane 2-complex X equipped with a semiflow \psi, and inspired by McMullen's Teichm\"uller polynomial for fibered hyperbolic 3-manifolds, we construct a polynomial invariant \m for (X,\psi) and investigate its properties. Specifically, \m determines a convex polyhedral cone \C_X in H^1(G;\R), a convex, real-analytic function \H:\C_X\to\R, and specializes to give an integral Laurent polynomial \m_u(\zeta) for each integral u\in\C_X. We show that \C_X is equal to the "cone of sections" of (X,\psi) (the convex hull of all cohomology classes dual to sections of of \psi), and that for each (compatible) cross section \Theta_u with first return map f_u:\Theta_u\to\Theta_u, the specialization \m_u(\zeta) encodes the characteristic polynomial of the transition matrix of f_u. More generally, for every class u\in\C_X there exists a geodesic metric d_u and a codimension-1 foliation \Omega_u of X transverse to \psi so that after reparametrizing the flow \psi^u_s maps leaves of \Omega_u to leaves via a local e^{s\H(u)}-homothety. Among other things, we additionally prove that \C_X is equal to (the cone over) the component of the BNS-invariant containing u_0 and that each primitive integral u\in\C_X induces a splitting of G as an ascending HNN-extension over a finite-rank free group along an injective endomorphism \phi_u. For any such splitting, we show that the stretch factor of \phi_u is exactly given by e^{\H(u)}. In particular, we see that \C_X and \H depend only on the group G and epimorphism u_0.

Using existing technology, we prove a Masur-Minsky style distance formula for flip- graph distance between two triangulations, expressed as a sum of the distances of the projections of these triangulations into arc graphs of the suitable subsurfaces of S.