12 problems

Entropy conjecture. The map $\varphi$ minimizes entropy in its homotopy class, and

Let $\Sigma$ be a surface, let $\operatorname{MCG}(\Sigma)$ denote its mapping class group, and let $S$ be a finite generating set for $\operatorname{MCG}(\Sigma)$. Write $d_S$ for…

Let $S_g$ be a closed orientable surface of genus $g>2$, and let $M(S_g)$ denote its mapping class group. A subgroup of $M(S_g)$ is normal all pseudo-Anosov if it is normal and eve…

AMU conjecture. For a mapping class $\phi\in\Gamma$, $\phi$ is Pseudo-Anosov or reducible with Pseudo-Anosov pieces if and only if $\rho_k(\phi)$ is infinite order for large enough…

Let $\phi$ be a pseudo-Anosov mapping class of a surface of arbitrary genus, and let $k$ denote the level of the associated TQFT representation. Infinite-order TQFT representation…

Let $\Sigma$ be a complete hyperbolic surface of finite type that is not a three-punctured sphere, and let $\operatorname{Mod}(\Sigma)$ be its mapping class group. Let $S$ be a spe…

Let $\Br_n$ be the braid group on $n$ strands, identified with the mapping class group of a disk with $n$ marked points relative to its boundary. A braid is pseudo-Anosov if its ma…

Fix $m\geq 2$, and let $f_1,f_2\in\operatorname{PA}(m)$ be $m$-modal standard PCP zig-zags of pseudo-Anosov type. Let $\lambda_i$ be the growth rate of $f_i$, and let…

A real number $bb>1$ is a bi-Perron unit if it is a positive real algebraic unit whose Galois conjugates lie in the annulus … except possibly for $bb$ and $bb^{-1}$. Bi-Perron suff…

Let $\beta=\beta^+\Delta$ be a positive pseudo-Anosov 4-braid containing a half-twist and whose closure is a knot. Let $\lambda_{R(\beta)}$ be the 1-cocycle spectral radius, and le…

Let $S$ be an orientable closed surface of genus $g \ge 1$, and denote its mapping class group by $\operatorname{MCG}(S)$. An element of $\operatorname{MCG}(S)$ is pseudo-Anosov (P…

Thurston's conjecture. The pseudo-Anosov dilatations are precisely the algebraic units that are Perron and larger than the Galois conjugates of their inverses.