6 problems

GHW decomposition conjecture. There exists such a decomposition $(\ast)$ in which $\Gamma_1$ and $\Gamma_2$ are GHW Bieberbach groups of dimension $n-1$.

Let the decompositions proved in the theorem, together with the known decompositions for parabolic (parahoric) subgroups in Tits theory, be formulated using the rings…

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…

One-endedness conjecture. Each vertex group $\mathcal{Y}(v)$ is one-ended.

Algorithmic central-decomposition conjecture. A combination of these results will produce a polynomial-time algorithm to find fully refined central decompositions of arbitrary fini…