23 problems

Let $\text{Teich}_g$ be the Teichmüller space of closed genus $g\geq 2$ Riemann surfaces. Let $h$ be any Riemannian metric, or any Finsler metric satisfying suitable weak regularit…

Let $M$ be a closed Riemannian manifold, let $\widetilde{M}$ be its universal cover, and let $\operatorname{Isom}(\widetilde{M})$ be its isometry group containing $\pi_1(M)$. Quant…

Let $M$ be a Riemannian manifold with universal cover $\widetilde{M}$, and let $\operatorname{Isom}(\widetilde{M})$ denote its isometry group. An orbibundle with locally symmetric…

Let $\mathbb U$ be Urysohn's universal metric space and let $Iso(\mathbb U)$ denote its isometry group. For $\varphi\in Iso(\mathbb U)$, write $Fix(\varphi)$ for the set of fixed p…

Finite-volume most-symmetry conjecture. Proposition 1 remains true when $M$ has finite volume and $h$ is any finite-volume Riemannian metric; that is, $\operatorname{Isom}(M,h)$ sh…

Let $(M_1,g_1,J_1,omega_1)$ and $(M_2,g_2,J_2,omega_2)$ be compact Kähler manifolds. Let $\operatorname{Isom_{\mathrm{Kah}}}(M_i)$ denote the Lie group of holomorphic isometries of…

Let $(M,g)$ be a 3-dimensional metric Lie group. A metric is complete when its geodesics are defined for all parameter values, and it has constant sectional curvature when every se…

Let $\overline{N}\cong \mathcal{H}\times L$ be the q-map space, with $g_{L}(p)$ the left-invariant metric on the fiber $L_p\cong L$ parametrized by $p\in\mathcal{H}$. Let $G$ be th…

Let $G$ be a $k$-step nilpotent Lie group with $k\geq 2$, and let its isometry Lie algebra and Lie algebra of first integrals be denoted by the corresponding Lie algebras. First-in…

Let $\mathcal{GH}$ be the metric space of isometry classes of compact metric spaces, endowed with the Gromov–Hausdorff metric. Triviality conjecture. The isometry group of…

Let $D$ be a cross-section of a proper open cone, and let $\operatorname{Isom}(D)$ and $\operatorname{Coll}(D)$ denote respectively the isometry group and the collineation group of…

Let $\Omega$ be an open bounded convex domain, and let a cone over $\Omega$ mean an open proper convex cone whose projectivization contains $\Omega$. A non-projective isometry is a…

Let $\Omega$ be an open bounded convex domain in an affine subspace of $\mathbb{RP}^n$. Write $Isom(\Omega)$ for its Hilbert-metric isometry group. de la Harpe's conjecture. The gr…

Lemmens–Walsh conjecture.

Let $G$ be a discrete group and let $X$ be a real Banach space with … An equivalent renorming of $X$ is a norm $|||\cdot|||$ equivalent to its original norm, and…

Farb and Weinberger's conjecture. 1. $M$ is aspherical, smoothly irreducible, $\pi_1(M)$ has no non-trivial normal abelian subgroup, and

Let $X$ be a complex HI space, meaning a hereditarily indecomposable Banach space, and let the group of isometries of $X$ act on $X$. Near-triviality conjecture. The group of isome…

Let $G$ be a group and let $X$ be a real Banach space with $\dim X \geqslant |G|$. An equivalent renorming of $X$ is a norm equivalent to its original norm, and…

Let $M_\infty$ be a limit space of Riemannian manifolds with a uniform lower Ricci curvature bound. Lie-group conjecture. The isometry group of any limit space $M_\infty$ is a Lie…

Symmetric-cone conjecture. The groups $\mathrm{Isom}(X)$ and $\mathrm{Coll}(X)$ differ if and only if the cone generated by $X$ is symmetric and not Lorentzian. In that case, the i…

de la Harpe's conjecture. The group $\mathrm{Isom}(X)$ is a Lie group, and $\mathrm{Isom}(X)$ acts transitively on $X$ if and only if $\mathrm{Coll}(X)$ acts transitively on $X$.

Let $S=S_{g,n}$ be a surface, let $d$ be a complete, finite-covolume, ${\rm Mod}(S)$-invariant Finsler metric on $\operatorname{Teich}(S)$, and suppose that $3g-3+n\geq 2$. Theorem…

Let $X$ be a Banach space and let $G$ be a group. A group is represented on $X$ if it is isomorphic to a group of linear isometries of some equivalent norm on $X$. Jarosz's represe…