16 problems

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…

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…

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.

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

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

Let $X$ be a spherical building, and let $Y$ be a convex contractible simplicial subcomplex of $X$. Let $H$ be an automorphism group of $X$ stabilizing $Y$. Tits' center conjecture…

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 $G$ 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…