Tilings of the plane, hyperbolic groups and small cancellation conditions (Q2753253)

It is considered a class of groups \(\mathcal T\) which satisfies a generalized \(C(4)\)-\(T(4)\) condition of small cancellation theory. To every group \(G\) in \(\mathcal T\) it is associated a finite set \(T(G)\) of colored squares in the plane, Wang prototiles. It is proved that the group \(G\in{\mathcal T}\) is hyperbolic iff \(T(G)\) is not a solvable set of prototiles, meaning one cannot tile the plane by copies of the prototiles in \(T(G)\) respecting the color-matching conditions and allowing only translation of the prototiles in \(T(G)\). The second result is that \(G\) has \(\mathbb{Z}\times\mathbb{Z}\) as a subgroup iff \(T(G)\) is a solvable periodic set of prototiles.