9 problems

Let $L$ be a flag complex, let $\mathbb{F}$ be a skew field, and let $\varphi\colon A_L\to\mathbb{Z}$ be an epimorphism. Suppose that $BB_L^\varphi$ is of type $\mathsf{F}$. Minima…

Volume-entropy conjecture. If

Let $(M,g)$ be a closed Riemannian $3$-manifold, let $R_g$ denote its scalar curvature, and let $h(g)$ denote its volume entropy. Agol–Storm–Thurston conjecture. If … then … This c…

For $-1\leq\alpha\leq 1$, let $G_\alpha$ be the homogeneous Riemannian manifold in the interpolating family, and let $h(G_\alpha)$ denote its volume entropy, defined by … where…

Let $(P_n)$ be a sequence of quasi-optimal pseudomanifolds representing $n\,\mathbf a$, and let $\varphi_n:\pi_1(P_n)\to\pi_1(X)$ be the induced homomorphism. Write…

Let $(P_n)$ be a sequence of quasi-optimal pseudomanifolds representing multiples of a homology class, equipped with the piecewise-flat metric in which every $m$-simplex is isometr…

Let $\Omega\subset\mathbb{R}\mathbb{P}^n$ be a convex proper open set, and let $h_{vol}(\Omega,d_{\Omega})$ denote the volume entropy of its Hilbert geometry, defined using the Bus…

Let $X$ be a closed oriented smooth Riemannian $4$-manifold, and let $\mu(X)$ denote its volume entropy. Assume that … and that $X$ admits a quasi-non-singular solution to the norm…

Let $K$ be an $n$-dimensional convex set equipped with its Hilbert geometry. Fix a volume and, for a metric ball $B(o,r)$, define the upper volume entropy by … The hyperbolic space…