10 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…

Thurston's conjecture. Given any pair of measured laminations $L^{+},L^{-}\in\mathcal{MGL}_{\pi}(S)$ that fill $S$, there is a unique quasi-Fuchsian manifold $Q$, up to isotopy, su…

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…

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 $\mathcal{PML}^+(S)$.

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…