9 problems

Let $W$ be a Coxeter group, let $\widetilde{S}$ be its smallest Garside shadow, and consider the automaton constructed from $\widetilde{S}$ that recognises the language of reduced…

Let $(G_{1,1},X)$ be the group–space pair whose patterns are being studied, and let a pattern be a sequence encoded by parenthesized blocks; within a block, $(0,-1)$ or $(1,0)$ den…

Parkinson–Yau's conjecture. The set $\Gamma$ is the smallest Garside shadow in $(W,S)$.

Let $W$ be a Coxeter group, let $\mathbb{T}$ denote its set of cone types, and let $\Gamma$ be the set of minimal-length representatives, called gates, of the parts of the cone typ…

Let $G$ act discretely, cocompactly and isometrically on a finite-dimensional hyperbolic space ${\mathbb H}_n$. The growth series of $G$ is the generating function that records the…

Strict-containment conjecture. The class of quasi-automatic semigroups strictly contains the class of asynchronously automatic semigroups.

Let … For $(m,n)\in\{(7,10),(7,11),(8,6),(12,4)\}$, the group $G$ is known to be infinite. Non-automaticity conjecture. In each of these four cases, $G$ is not automatic and theref…

Let $G$ be a word-hyperbolic group. An automatic structure for $G$ has property R when its accepting automaton is both undirected and Perron–Frobenius, meaning that the adjacency m…

Shephard groups' bi-automaticity conjecture. Every Shephard group is bi-automatic. The conjecture extends analogous conjectures for Artin and Coxeter groups. The paper proves it fo…