25 problems

Let $G$ be a group acting by type-preserving simplicial isometries on a building $\Delta$. Let $X$ be the Davis realization of $\Delta$, and call the action locally elliptic if every…

Let ${\mathcal G}_k$ be the Grassmannian of $k$-dimensional singular subspaces of a building of type $\textsf{X}_{n}=\textsf{C}_{n},\textsf{D}_{n}$, and let ${\mathfrak G}_k$ be it…

Unfoldability conjecture. A billiard triangle in $\Delta$ with sides $\overline{ox_0}$, $\overline{ox_k}$ and $x_0x_1\ldots x_{k-1}x_k$, and with the special vertices $0,x_0,x_k$,…

Cocycle-expansion conjecture. For every such lattice $\Gamma$,

Lee–Szczarba conjecture. For a prime $p$ and $n\geq 2$, the induced map

Let a building of type $\widetilde{A}_n$ be a simplicial complex built from Coxeter complexes of type $\widetilde{A}_n$, and let its 1-skeleton be the graph whose vertices and edge…

Marquis's conjecture. The group $G$ has a global fixed point in $X\cup\partial X$.

Let $q$ be the parameter in the construction of $X^n_\varnothing$, let $0\leq i\leq n-1$, and let $\Delta$ act on $C^i(X^n_\varnothing)$. The positive eigenvalues of $\Delta$ are a…

Let $B_{i,\infty}^\pm$ be the stabiliser of the corresponding end, with subgroups $B_W^\pm$, $U_i$, and $W^{even}$ as defined in the paper. Structure decomposition conjecture. For…

Let $A^7=g\cdot A^7$ for some $g\in B_{i,\infty}^7$, where $A^7$ is the standard apartment and $U_i$ is the corresponding unipotent subgroup. Transitivity conjecture. For each such…

Pre-building Finsler isometric-embedding conjecture. The factorization $f:B^{\rm pre}_{\phi}\rightarrow B$ is an isometric embedding for the Finsler distance.

Bounded-region finiteness conjecture. On bounded regions, the process stabilizes in finitely many steps.

No-BPS finite stabilization conjecture. The sequence $Z_i$ is well-defined until it reaches a construction with no more fourfold points or sixfold points with triple fold lines; th…

Folding-in versality conjecture. Every such map $F\rightarrow B$ extends to a map $G\rightarrow B$, although the extension need not be unique.

Building realization conjecture. The topological space $F(p)$ has a natural structure of an $\mathbb{R}$-building modeled on the affine space $A$.

Topological structure conjecture. The topological space $F(p)$ is Hausdorff and is a CW-complex.

Let $G$ be a semisimple complex Lie group, let $X$ be a Riemann surface, and let … be a point in the corresponding Hitchin base. Let $h^{\phi}$ be the harmonic map to the universal…

Let $X$ be a Riemann surface and let $\phi$ be a spectral cover. Let $\mathcal W$ be the spectral network associated to $\phi$, and let $\widetilde{\mathcal W}$ be its lift to…

Let $X$ be a Riemann surface and let $\phi$ be a smooth spectral cover of $X$. A universal harmonic $\phi$-map is a harmonic $\phi$-map $h^{\phi}:\widetilde X\to\mathcal B^{\phi}$…

Let $\mathcal G$ be a Chevalley group of adjoint type $\tau$, rank $r$, and dimension $n=r+2m$, with Lie algebra $\mathfrak g$. Let $\mathbb P[\Psi]$ be the associated…

A finite thick building is a finite building in which every panel is contained in at least three chambers, and its dimension is denoted by $d$. Non-embeddability conjecture. No $d$…

Let $G$ be an almost connected locally compact group. Recall that a group has property $\operatorname{(FB)}$ when every measurable action of $G$ on a finite-rank building stabilise…

Caprace–Marquis weaker conjecture. Every split spherical $BN$-pair of $G(k)$ is virtually trivial.

Caprace–Marquis conjecture. Every split spherical $BN$-pair of $G(k)$ is trivial.

Let $(W,S)$ be an irreducible non-spherical and non-affine Coxeter system with $S$ finite, let $X$ be a building of type $(W,S)$, and let $G$ be a group acting on $X$ by automorphi…