7 problems

Let $G$ be a connected semi-simple Lie group with no compact simple factor, and suppose that the Lie algebra of $G$ has $\mathbb{R}$-rank at least $2$. Let $\Gamma$ be an irreducib…

Unique irreducible composition conjecture. Every planar median graph $G$ has a unique composition

A group is cubulable if it acts properly and cocompactly on a median graph. For $n\geq 1$, $(\mathrm{FW}_n)$ denotes the fixed-point property for actions on median graphs of cubica…

Zhang et al.'s conjecture. The cube polynomial $C(G,x)$ is unimodal.

Let $G_{d,q}$ be the covering graph of the face-lattice of the finite projective space $\operatorname{PG}(d,q)$ of dimension $d$ and order $q$, and let $p(G_{d,q})$ denote the conn…

Median braid-graph conjecture. Every such braid graph is a median graph.

Retraction characterization of hyper-median partial cubes. A partial cube is hyper-median if and only if it is a retract of a Cartesian product of bipartite cellular graphs.