14 problems

Let $G$ be a finitely generated group. The Ballier–Stein domino problem conjecture. The domino problem for $G$ is decidable if and only if $G$ is virtually free. This conjecture pr…

Let $G$ be a finitely generated group. A group is virtually free if it has a finite-index free subgroup, and its domino problem is the decision problem for finite local colouring c…

Let $G$ be a finitely generated group. The domino problem asks whether, given a finite-type constraint system (equivalently, a $G$-subshift of finite type), there exists a configur…

Seeded domino conjecture. The problem $\textnormal{SDP}(G)$ is decidable if and only if $G$ is virtually free.

The Domino Conjecture. The problem $\textnormal{DP}(G)$ is decidable if and only if $G$ is virtually free.

Seeded and recurring domino conjecture. The seeded and recurring domino problems are decidable on exactly the virtually free groups.

Virtually free domino-problem conjecture. Virtually free groups are precisely the groups with decidable domino problem.

For integers $p,q\geq 2$, let ${\mathsf{DL}(p,q)}$ be the Diestel–Leader graph, and let the unlabelled graph mean this graph considered without its standard edge labelling. A seede…

Let $G$ be a one-ended word-hyperbolic group, meaning a word-hyperbolic group with one end. Let the surface group of genus $2$ be the fundamental group of a closed orientable surfa…

Let $G$ be a finitely generated group. The domino problem of $G$ asks whether there is an algorithm deciding, for every finite set of forbidden patterns, whether a configuration av…

Weak-aperiodicity conjecture. $G$ has no weakly aperiodic SFT if and only if it is virtually cyclic.

Domino-problem conjecture. $G$ has a decidable domino problem if and only if it is virtually free.

Let $G$ be a finitely generated group. The domino problem for $G$ asks whether a given finite collection of local forbidden patterns admits a global configuration on a Cayley graph…

The domino problem concerns whether a finite set of local tile constraints admits a global tiling of a graph or group. The domino problem on the hyperbolic plane is the correspondi…