  We define a quasi-Fatou component of a quasiregular map as a connected
component of the complement of the Julia set. A domain in $\mathbb{R}^d$ is
called hollow if it has a bounded complementary component. We show that for
each $d \geq 2$ there exists a quasiregular map of transcendental type $f:
\mathbb{R}^d \to \mathbb{R}^d$ with a quasi-Fatou component which is hollow.
  Suppose that $U$ is a hollow quasi-Fatou component of a quasiregular map of
transcendental type. We show that if $U$ is bounded, then $U$ has many
properties in common with a multiply connected Fatou component of a
transcendental entire function. On the other hand, we show that if $U$ is not
bounded, then it is completely invariant and has no unbounded boundary
components. We show that this situation occurs if $J(f)$ has an isolated point,
or if $J(f)$ is not equal to the boundary of the fast escaping set. Finally, we
deduce that if $J(f)$ has a bounded component, then all components of $J(f)$
are bounded.
