4 problems

COP-Farrell-Jones conjecture for Hecke algebras. For every uniformly regular ring $R$ with $\mathbb{Q} \subseteq R$, the projection

Let $G$ be a compactly generated, topologically simple totally disconnected locally compact group, let $x\in G$ satisfy $s(x^{-1})>1$, and let $V_{--}\rtimes\langle x\rangle$ be th…

Flat-rank one subgroup conjecture. Every flat subgroup $H$ of flat-rank $1$ in $G$ has a limit set containing two elements, both fixed by $H$. Moreover, $H_1$ is relatively compact…

Directions and boundary conjecture. The map that assigns to each element of non-trivial scale its attracting boundary point defines an injection from the set of directions of $G$ i…