  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 $c \in (0,1)$.
