8 problems

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 $\Sigma$ 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 $\Br_n$ 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…

Let $S_{g,n}$ be an orientable surface of genus $g$ with $n$ punctures, and let $f\in\mathrm{Mod}(S_{g,n})$ be a pseudo-Anosov mapping class, meaning that it has a representative p…

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…