15 problems

Let $S$ be a surface, let $\operatorname{Mod}(S)$ be its mapping class group, and let $\Lambda_G$ denote the limit set of a subgroup $G\leq \operatorname{Mod}(S)$ in the projective…

Generic spectrum conjecture. For a generic $\rho: \Sigma_g \longrightarrow \mathrm{SL}(2,\mathbb{R})$ which is not simple Fuchsian, the spectrum of $\rho$ is closed and has empty i…

Let $M$ be the surface under consideration and let $\lambda$ be an arbitrary oriented lamination on $M$ with a transverse measure. Let $\tilde v:\tilde M\to\mathbb R$ be a primitiv…

Let $M$ be as in Theorem, let $\mathcal{M}\mathcal{L}_0(M)$ be the space of measured laminations without boundary, and let $\overline\chi:\mathcal{M}\mathcal{L}_0(M)\to\mathbb R$ b…

Let $M$ be as in Theorem, let $\mathcal{M}\mathcal{L}_0(M)$ be the space of measured laminations without boundary in $M$, and let $\mathcal{P}_{\mathcal T}$ be the relevant polyhed…

Let $g$ be the genus, let $\kappa$ be a partition of $4g-4$, and let $\cI(\kappa)\subset \ML\times \cT_g$ consist of pairs whose complementary regions are symmetric ideal hyperboli…

Let $\ML_0$ denote the subspace of measured laminations with the property used to define the semi-conjugacy, and let $\cT_g$ be Teichmüller space. Continuity conjecture. The semi-c…

Closure-equality conjecture. The inclusions in Theorems and are equalities, except when $\lambda$ is the bounding geodesic of the genus-three closed surface.

Rational-density conjecture.

Sullivan-type invariant-measure conjecture. There exists a unique, up to scaling, ergodic $\operatorname{Map}(S)$-invariant Radon measure supported on $\mathcal{ML}^+(S)$, and it i…

Limit-set conjecture. The limit set of $\operatorname{Map}(S)$ in $\mathcal{PML}(S)$ is $\mathcal{PML}^+(S)$.

Let $ctheta_+$ and $ctheta_-$ be filling measured laminations on a closed surface. Suppose there exist hyperbolic surfaces $crho_L$ and $crho_R$ such that $crho_R$ is obtained from…

Let Teichmüller space be the space of marked conformal structures on a surface, and let its Bers boundary be the boundary arising from a Bers embedding. Let unmeasured lamination s…

Tightness conjecture. The automorphism $f$ is efficient if and only if its associated pair of $f$-invariant laminations is tight.

Let $T\in\partial cv_N$ be an $\mathbb R$-tree with dense orbits, and suppose that its dual lamination $L(T)$ is uniquely ergodic; then $T$ is dually uniquely ergodic. In this case…