25 problems

Tits Centre Conjecture. If $\Sigma$ is a convex non-completely-reducible subcomplex of $\Delta$, then $\Sigma$ has a simplicial $\Gamma$-centre.

A convex subcomplex of a spherical building is a subcomplex containing every apartment-geodesic segment between any two of its points. An automorphism preserves a subcomplex when i…

Conjecture. Either $\frak G$ is a Moufang quadrangle, or $\frak G$ is of Clifford type with topological parameters $(3,4k)$ or $(7,8)$.

Let $\Delta$ be a spherical building, and let $\sigma\in\Delta$ be any non-empty simplex. Let ${\mathcal{OPD}}(\Delta)$ and ${\mathcal{PD}}(\Delta)$ denote the ordered partial deco…

Let $G$ be an arbitrary connected reductive group, let $k$ be a field, and let $\Delta(G,k)$ be its spherical building. Let ${\mathcal{OD}}(\Delta(G,k))$ denote the ordered decompo…

Let $X_{n-2,q}$ be the spherical building of dimension $n-2$ and thickness $q+1$. For $0\leq i\leq n-2$, let $d_i$ and $\delta_i$ denote the coboundary and boundary operators at le…

Donald's spherical building conjecture. Donaldson conjectured that the geometry of the space of Kähler metrics should admit an infinite-dimensional spherical building structure, as…

Tits's center conjecture. Every convex subset $S$ of chambers in $\Delta$ must either contain a pair of opposite chambers, or $\mathrm{Stab}_G(S)$ fixes a non-empty partial flag of…

Suppose that $(\widetilde G_{k^{\mathrm a}},\Gamma_{k^{\mathrm a}})$ is quasisemisimple. Write $G^{\mathrm{red}}$ for the maximal pseudo-reductive quotient of $G$, which is reducti…

Ovoid fix-structure conjecture. No such ovoid is the fix structure of any collineation.

Strong Tits Centre Conjecture for vector edifices. If $C$ is a closed convex non-completely-reducible cone in $V_G({\mathbb K})$, then $C$ has an unopposed $\Gamma$-centre.

Strong Tits Centre Conjecture. If $\Sigma$ is a closed convex non-completely-reducible subset of $\Delta_G({\mathbb K})$, then $\Sigma$ has an unopposed $\Gamma$-centre.

A spherical building is a simplicial complex with the spherical building structure, and its coboundary expansion is the corresponding expansion parameter for cochains. Let $d$ be a…

Let $X$ be a spherical building, and let $Y$ be a convex contractible simplicial subcomplex of $X$. Let $H$ be a subgroup of the automorphism group of $X$ that stabilizes $Y$. Tits…

E8(2) uncapped-automorphism conjecture. Let

Involution opposition-diagram conjecture. In each of these cases, there exists an involution with the given opposition diagram, but it is never conjugate to any Galois involution.

Let $\Delta$ be an infinite thick irreducible compact totally disconnected spherical building of rank $2$. Moufang property conjecture. If $\Delta$ is strongly transitive, then…

Let $\Delta$ be an infinite thick irreducible compact spherical building of rank at least $2$. Strong transitivity characterization conjecture. The building $\Delta$ is the spheric…

Bruhat–Tits boundary conjecture. Then $B$ is the Tits boundary of a locally finite simplicial Bruhat–Tits building.

Center conjecture. At least one of the following two possibilities holds:

Let $B$ be a spherical building and let $C\subseteq B$ be a closed convex subset. Write $rad_C(C)$ for the intrinsic radius of $C$, namely the infimum of the radii of balls centere…

Let $B$ be a spherical building and let $C\subseteq B$ be a closed convex subset. Fixed-point conjecture. Either $C$ is a subbuilding, or the action … has a fixed point. This gener…

Let $B$ be a spherical building and let $K\subseteq B$ be a convex subcomplex. A center is a point fixed by the automorphisms of $B$ preserving $K$. Tits' Center Conjecture. Either…

Let $B$ be a spherical building and let $K\subseteq B$ be a convex subcomplex. Center Conjecture. Then $K$ is a subbuilding or the action … of the automorphisms of $B$ preserving…

Let $G$ be a connected group, let $X=X(G,k)$ be its spherical Tits building over $k$, and identify $X$ with its geometric realization. A subcomplex $Y$ of $X$ is convex if, wheneve…