11 problems

Let $G$ be a lattice in a higher-rank semisimple Lie group, and consider the asymptotic cones of $G$. Drutu–Mozes–Sapir conjecture. $G$ does not have cut-points in any of its asymp…

Let $G$ and $G'$ be groups with Cayley graphs whose asymptotic cones are isometric. A group is non-elementary hyperbolic if it is hyperbolic and is not virtually cyclic. Sapir's co…

Fix a polystable negative valuation $v$ on $X$. Let $\mathcal M_v$ be the space of compatible Calabi–Yau metrics on $X$ with $v=v_\omega$, let $\mathcal C_v$ be the corresponding w…

Let $X$ be a complete Calabi–Yau manifold in the setting of Theorem … should hold without the quadratic curvature decay assumption. This is the expected extension of the paper's ma…

Restrict to the free group $G=F(A)$, let $\Theta\subseteq P(G)$ be a set of cyclically reduced representatives for the equivalence classes in $P(G)$, and let $\tau$ be the stable n…

Let $G$ be a group with norm $\|\cdot\|$, and let $F(G)$ denote the free group on the underlying set of $G$, with generators written $\overline g$. For $g_1,\dots,g_n\in G$, compar…

Let $G$ be a group equipped with the bi-invariant metric norm $\\|\cdot\\|$. For elements $g_1,\dots,g_n\in G$, consider the sequence of normalized norms of products of equal power…

Let $\Gamma$ be a group and let $C$ be any asymptotic cone of $\Gamma$. Gromov's asymptotic-cone fundamental-group conjecture. The fundamental group $\pi_1(C)$ is either trivial, e…

Let $\Gamma$ be a finitely generated nilpotent group, and let $r$ denote its nilpotency class. The rate of convergence of its rescaled word metric to the asymptotic cone is measure…

A graph is called $delta$-CAT(0) when its geodesic triangles satisfy the corresponding CAT(0) comparison condition up to an additive constant, and it is asymptotically CAT(0) when…

Sapir's conjecture. Under these hypotheses, the critical exponents of the two groups should coincide.