23 problems

Virtual Betti number conjecture. One has

Let $\Sigma$ be an integral homology sphere, let $\tau:\Sigma\to\Sigma$ be an orientation-preserving diffeomorphism, and let $X$ be its mapping torus. Let $\lambda_{FO}(X)$ be the…

Let $\Sigma$ be an integral homology sphere, let $\tau:\Sigma\to\Sigma$ be an orientation-preserving diffeomorphism, and let $X$ be its mapping torus. Let $\lambda_{FO}(X)$ be the…

Let $\mathbb{F}$ be a finitely generated free group and let $\psi\to \mathbb{F}$ be a fully irreducible monomorphism, meaning that no proper nontrivial finitely generated free fact…

Density conjecture. If $\mathcal{S}(\Sigma)$ is non-empty, then $\mathfrak{F}(\Sigma)$ is dense in $\mathcal{M}(\Sigma)$.

Let $F_n$ be a free group of rank $n$, let $Gamma=F_n\rtimes_{alpha}langle trangle$ be a unipotent polynomial suspension, and let $F_m$ be a free group of rank $m$. Suppose that…

Let $S$ be an oriented surface with negative Euler characteristic, let $\varphi$ be a periodic diffeomorphism, and let $[\gamma]$ be a $\varphi$-invariant smooth character with…

Surface curvatures of mapping tori of free group endomorphisms. Then $\sigma_+(X)<0$ unless $\pi_1(X)$ has a Baumslag--Solitar subgroup.

Irreducible curvatures of mapping tori of free group endomorphisms. Then $\rho_+(X)=0$.

Let $\Sigma$ be a compact orientable surface and let $f\in\mathrm{Mod}(\Sigma)$. Let $M_f$ denote the mapping torus of $f$, and let $\lVert M_f\rVert$ denote its Gromov norm. A com…

Let $M\in\operatorname{SL}(2,\mathbb{Z})$ have a decomposition into $S$ and $T$ generators, and let…

Li–Ni's conjecture. If $M$ is a closed oriented reducible $3$-manifold, then

Let $\mathcal{A}$ be a dg category with strictly commuting strict dg auto-equivalences $\phi$ and $\psi$, and let $M_{\phi,\psi}$ be the associated doubly twisted mapping torus cat…

Let $\mathcal{A}$ be as in Theorem. Let $\phi$ and $\phi'$ be auto-equivalences satisfying the stated conditions, and let $M_\phi$ and $M_{\phi'}$ be their mapping torus categories…

Let $\Sigma$ be a compact oriented surface and let $f\in\mathrm{Mod}(\Sigma)$ be pseudo-Anosov. Let … be the mapping torus of $f$, and let $TV_r(M_f)$ denote its $SO(3)$ Turaev–Vir…

Let $\Sigma$ be a surface and let $\varphi$ be a pseudo-Anosov map such that the mapping torus $(\Sigma\times S^1)_{\varphi}$ is hyperbolic. Let…

Fix a generating set $\Gamma$ of the mapping class group of a surface of genus $g$. Let $\phi_N$ be a random word of length $N$, and let $V(\phi_N)$ be the volume of its mapping to…

Consider random mapping tori whose monodromies are random words, and let the genus of the fiber tend to infinity. Let the expected growth rates refer to growth as a function of the…

Let $N$ denote the length of a random monodromy word and let $N^3$ be the associated mapping torus. Write $\lambda_1(N)$ for the bottom eigenvalue of the Laplacian. Asymptotic eige…

Let $S$ be a surface, and let $N$ denote the length of a random monodromy word defining a mapping torus of $S$. The injectivity radius is the radius of the largest embedded metric…

Symplectic reducible-fiber conjecture. Then $Y=S^2\times S^1$ and

Torsion conjectures. The manifold $T$ has exponential torsion homology growth with respect to some tower of finite covers. In fact, $T$ has exponential torsion homology growth with…

A mapping torus of a free group automorphism is the ascending HNN extension … where $F_k$ is a free group and $\phi$ is an automorphism of $F_k$. Virtual mapping-torus conjecture.…