We consider three-dimensional domino tilings of cylinders \(\mathcal {R}_N = \mathcal {D}\times [0,N]\) where \(\mathcal {D}\subset \mathbb {R}^2\) is a fixed quadriculated disk and \(N \in \mathbb {N}\) . A domino is a \(2 \times 1 \times 1\) brick. A flip is a local move in the space of tilings \(\mathcal {T}(\mathcal {R}_N)\) : remove two adjacent dominoes and place them back after a rotation. The twist is a flip invariant which associates an integer number to each tiling. For some disks \(\mathcal {D}\) , called regular, two tilings of \(\mathcal {R}_N\) with the same twist can be joined by a sequence of flips once we add vertical space to the cylinder. We have that if \(\mathcal {D}\) is regular then the size of the largest connected component under flips of \(\mathcal {T}(\mathcal {R}_N)\) is \(\Theta (N^{-\frac{1}{2}}|\mathcal {T}(\mathcal {R}_N)|)\) . The domino group \(G_\mathcal {D}\) captures information of the space of tilings. A disk \(\mathcal {D}\) is regular if and only if \(G_{\mathcal {D}}\) is isomorphic to \(\mathbb {Z} \oplus \mathbb {Z}/(2)\) ; sufficiently large rectangles are regular. We prove that certain families of disks are irregular. We show that the existence of a bottleneck in a disk \(\mathcal {D}\) often implies irregularity. In many, but not all, of these cases, we also prove that \(\mathcal {D}\) is strongly irregular, i.e., that there exists a surjective homomorphism from \(G_{\mathcal {D}}^+\) (a subgroup of index two of \(G_{\mathcal {D}}\) ) to the free group of rank two. Moreover, we show that if \(\mathcal {D}\) is strongly irregular then the cardinality of the largest connected component under flips of \(\mathcal {T}(\mathcal {R}_N)\) is \(O(c^N |\mathcal {T}(\mathcal {R}_N)|)\) for some \(c \in (0,1)\) .
