20 problems

Let $\Lambda$ be a graph, and let … be its associated right-angled Coxeter group. A finitely generated group has a quasi-isometry invariant Morse boundary, denoted…

Flag-complex partial-link conjecture.

Finite flag subcomplex vanishing conjecture.

Let $\mathcal{K}$ be a flag complex, let $L_\mathcal{K}$ be the graph Lie algebra over $\mathbb{Z}_2$, and let $N_\mathcal{K}$ be the Lie subalgebra generated by the Grbić–Panov–Th…

Polylogarithmic non-giant components conjecture. Suppose

Polylogarithmic second-component conjecture. Already at the emergence of a giant square component, the second largest square component should have support of only polylogarithmic o…

Let $\mathcal{G}_{n,p}$ denote the Erdős–Rényi random graph, and let $\Gamma\sim\mathcal{G}_{n,p}$. For a group or graph, “thick of order at most $k$” is used in the sense of the p…

Let $\mathcal{G}_{n,p}$ denote the Erdős–Rényi random graph, let $\Gamma\sim\mathcal{G}_{n,p}$, and let $W_{\Gamma}$ be the right-angled Coxeter group with presentation graph…

Let $W_{\Gamma}$ be a right-angled Coxeter group. A cubical coarse median rigidity conjecture. 1. There are only finitely many cubical coarse median structures on $W_{\Gamma}$. 2.…

Connectivity conjecture. For every $\epsilon>0$, if

Let $d \in \mathbb{N}$ and let (\Delta_n)_{n \in \mathbb{N}) be a sequence of finite $d$-dimensional buildings whose thickness satisfies … as $n \rightarrow \infty$. For each bui…

Let $(W,S)$ be a right-angled Coxeter group of finite rank. For $s,t\in S$, write $o(st)$ for the order of $st$, and let $N(u)$ denote the set associated to $u$ by the Coxeter grap…

Let $\Gamma$ be a random graph in $G(n,p(n))$, and let $\mathcal{CFS}$ denote the graph property used to characterize quadratic divergence of the associated right-angled Coxeter gr…

Let $W_\Gamma$ be a right-angled Coxeter group, and let $n$ denote its hypergraph index. Let thickness of order $n$ and divergence $x^{n+1}$ have their standard meanings for the gr…

Let $\Gamma$ be the defining graph of a right-angled Coxeter group $G_\Gamma$. An induced loop $\sigma$ in $\Gamma$ has length greater than $4$ and its vertex set contains no pair…

Let $\Gamma$ be a simplicial graph and let $W_{\Gamma}$ be the corresponding right-angled Coxeter group. Fix an order $n$. Equivalence conjecture. The following are equivalent:…

Let $n$-vertex graphs be sampled from a random graph model with density greater than $1/n$ and bounded away from $1$, and let the associated right-angled Coxeter group be the group…

Let $T$ and $T'$ be trees. Assume that they are simultaneously co-spectral, their complements are co-spectral, and their line graphs are co-spectral. Geodesic-growth conjecture. Th…

A geometric action on a CAT(0) space is a properly discontinuous, cocompact action by isometries. Let $X$ be a CAT(0) space, and consider right-angled Coxeter groups that act geome…

Let $(W,S)$ be a right-angled Coxeter system, with presentation diagram given by the defining commutation relations, and let the elementary separation condition referred to in the…