45 problems

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…

A bi-Perron number is an algebraic integer $\beta>1$ whose Galois conjugates all lie in the annulus $\beta^{-1}<|z|<\beta$, with $\beta^{-1}$ also among its conjugates. Fried's con…

Let $\varphi$ be a pseudo-Anosov homeomorphism of a punctured surface, and let $M_\varphi$ be its hyperbolic mapping torus. The Bonahon–Wong–Yang invariants are defined at roots of…

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

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…

Odd-genus normalized liminf conjecture.

Magic-manifold minimal-dilatation conjecture.

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…

The generalized pseudo-Anosov maps under consideration preserve a pair of measured foliations with singularities, denoted by ${\mathcal{F}}^{u,s}$. A map has finitely many singular…

Let $\rho:S\rightarrow S$ be a pseudo-Anosov map with expansion factor $\beta$, and let $\phi_t^u$ denote the unit-speed flow in the unstable direction. If $\beta$ is Pisot of degr…

Let $\varphi$ be a pseudo-Anosov homeomorphism of a punctured surface, and let $M_\varphi$ be its mapping torus with canonical longitude $\lambda$. For a complex root of unity…

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…

Let $\delta_n$ be the minimum dilatation among pseudo-Anosov braids with $n$ strands, and let $\underline{\delta}_n$ denote the referenced candidate value. The remaining values of…

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…

Let $\Sigma$ be the surface under consideration and let $\phi\in\operatorname{Mod}(\Sigma)$ be pseudo-Anosov. A finite cover $\widetilde{\Sigma}\to\Sigma$ is allowed to carry a lif…

Minimal accumulation point conjecture. The minimal accumulation point of this set is $\sigma^2$.

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

Let $f$ be a measurable pseudo-Anosov map, and let $h_{\mathrm{metric}}(f)$ and $h_{\mathrm{topological}}(f)$ denote its metric and topological entropies. Let $\lambda>1$ be the ex…

A measurable pseudo-Anosov map is the type of dynamical system described in the source. A 0-entropy quotient is the quotient associated to a surface diffeomorphism; a homoclinic cl…

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…

Let $f:S\to S$ be a fully-punctured pseudo-Anosov map and consider the veering triangulation of its mapping torus. A double hook circuit is the pair of hook circuits described in t…

McMullen's conjecture. Every pseudo-Anosov automorphism $f\colon S\to S$ admits a lift $f'\colon S'\to S'$ to some finite cover $S'\to S$ such that…

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