22 problems

For a finite type surface $S$, let $f\colon S\to S$ be a pseudo-Anosov homeomorphism, and let $(X,q)$ be the associated complex structure and holomorphic quadratic differential. Ca…

Fix a finite generating set for ${\rm Mod}_g$, let $B(r)$ be the ball of radius $r$, and define the density of $S\subset {\rm Mod}_g$ by … Let ${\cal P}$ be the set of pseudo-Anoso…

Let $S_{g,p}$ be a genus-$g$ surface with $p$ punctures, let $\brun(S_{g,p})$ denote the set of Brunnian pseudo-Anosov mapping classes on $S_{g,p}$, and let $L(\brun(S_{g,p}))$ den…

Let $\Sigma_{g,n}$ be a compact oriented surface, and let … be its mapping class group, with homeomorphisms and isotopies fixed on the boundary. For an odd integer $r\geq 3$, let…

For each genus $g$, let $\delta_{g,0}$ be the minimum dilatation of pseudo-Anosov maps on the closed surface of genus $g$. Let $\mu=\frac{1+\sqrt{5}}{2}\approx1.618$ be the golden…

Let $\underline{\delta}_9$ be the stated candidate value for the minimum dilatation of pseudo-Anosov braids with 9 strands, and let $x$ denote the variable in its defining polynomi…

Linear growth conjecture. The entropy of braids in the sequence $\\{\beta^{mk+1}\\}_{m=M+1}^\infty$, a lower bound on the knot genus of…

Exact minimal stretch factor conjecture. The normalized stretch factor of $f$ is greater than or equal to the largest real root of

For each genus $g$, let $\delta_{g,0}$ denote the minimum dilatation among pseudo-Anosov maps on the closed surface of genus $g$, and let $\mu$ be the golden ratio. Hironaka's gold…

Odd-genus Penner dilatation limit conjecture. The limit is the largest real root of

Let $S_g$ be a closed orientable surface, let $f\in\operatorname{Mod}(S_g)$ be pseudo-Anosov with stretch factor $\lambda$, and let $C$ be the constant appearing in the small-stret…

Odd-genus normalized liminf conjecture.

Odd-genus normalized limit conjecture.

Even-odd monotonicity conjecture. For all $k\geq 1$,

Odd-genus orientation-reversing stretch factor conjecture. For all $k\geq 2$, $\delta^+_{\mathrm{rev}}(S_{2k-1})$ is the largest root of

Genus 9 and 11 minimal stretch factor conjecture. The minimal stretch factors and minimal polynomials are predicted to be

Let $h$ be a pseudo-Anosov mapping class and let $\operatorname{\omega}(h)$ denote its spectral ratio. For a surface and an infinite family of pseudo-Anosov mapping classes, a P…

Let $(\Lambda,[I],K)$ be the triple corresponding to a pseudo-Anosov map $\phi\in Mod~(X)$, and let $\lambda_\phi$ be its dilatation. The integral order $\Lambda$ has class number…

Let $G=\operatorname{SU}(N)$, let $\Sigma_g$ be a closed surface of genus $g$, and let $\varphi\in\Gamma_g$ be a mapping class with stretch factor $\lambda$. Write $T_{\varphi^n}$…

N(1)-fibration asymptotic conjecture.

Magic-manifold minimal-dilatation conjecture.

Let $S_{g,n}$ be a surface of genus $g$ with $n$ punctures, let $\operatorname{Mod}(S_{g,n})$ be its mapping class group, and let $L_{\mathcal C}(\operatorname{Mod}(S_{g,n}))$ deno…