Laurent Bartholdi's paper addresses the domino problem for non-virtually-free hyperbolic groups. The domino problem asks whether an algorithm can determine if a given set of dominoes can edge-cover the Cayley graph of a group, with colors matching at vertices. The paper proves that for such groups, no such algorithm exists, answering a conjecture by Aubrun, Barbieri, and Moutot and contributing to the long-standing Ballier and Stein conjecture. The paper begins by defining the domino problem and its decidability properties, noting that if a group has a finite-index free subgroup, the problem is decidable. It then reviews known results, including the undecidability of the problem on certain groups like $\mathbb{Z}^2$ and groups containing $\mathbb{Z}^2$ as a subgroup. The author introduces the concept of translation-like actions and shows that if a group admits such an action, the domino problem is undecidable. The main result, Theorem A, states that for a non-virtually-free word hyperbolic group, the domino problem is decidable if and only if the group is virtually free. The proof relies on a criterion (Theorem 3.1) that guarantees undecidability of the domino problem on a graph if it can simulate a sequence of connected, infinite, amenable graphs with specific contraction maps. The paper then provides a detailed proof of Theorem 3.1, using a reduction from a problem on piecewise affine maps in the plane. It constructs a set of domino tiles and demonstrates how to construct solutions from orbits of the map, and conversely, how solutions can be used to find immortal points of the map. Finally, the paper applies these results to non-virtually-free word hyperbolic groups, showing that they can simulate a 2-regular tower of infinite, amenable, connected, bounded-degree graphs, thus proving Theorem A. The proof involves using a subshift of finite type constructed by Cohen, Goodman-Strauss, and Rieck, which encodes the relevant properties of the group's Cayley graph and its horosphere graphs.Laurent Bartholdi's paper addresses the domino problem for non-virtually-free hyperbolic groups. The domino problem asks whether an algorithm can determine if a given set of dominoes can edge-cover the Cayley graph of a group, with colors matching at vertices. The paper proves that for such groups, no such algorithm exists, answering a conjecture by Aubrun, Barbieri, and Moutot and contributing to the long-standing Ballier and Stein conjecture. The paper begins by defining the domino problem and its decidability properties, noting that if a group has a finite-index free subgroup, the problem is decidable. It then reviews known results, including the undecidability of the problem on certain groups like $\mathbb{Z}^2$ and groups containing $\mathbb{Z}^2$ as a subgroup. The author introduces the concept of translation-like actions and shows that if a group admits such an action, the domino problem is undecidable. The main result, Theorem A, states that for a non-virtually-free word hyperbolic group, the domino problem is decidable if and only if the group is virtually free. The proof relies on a criterion (Theorem 3.1) that guarantees undecidability of the domino problem on a graph if it can simulate a sequence of connected, infinite, amenable graphs with specific contraction maps. The paper then provides a detailed proof of Theorem 3.1, using a reduction from a problem on piecewise affine maps in the plane. It constructs a set of domino tiles and demonstrates how to construct solutions from orbits of the map, and conversely, how solutions can be used to find immortal points of the map. Finally, the paper applies these results to non-virtually-free word hyperbolic groups, showing that they can simulate a 2-regular tower of infinite, amenable, connected, bounded-degree graphs, thus proving Theorem A. The proof involves using a subshift of finite type constructed by Cohen, Goodman-Strauss, and Rieck, which encodes the relevant properties of the group's Cayley graph and its horosphere graphs.