  We show that for a number of parameterized problems for which only $2^{O(k)}
n^{O(1)}$ time algorithms are known on general graphs, subexponential
parameterized algorithms with running time $2^{O(k^{1-\frac{1}{1+\delta}}
\log^2 k)} n^{O(1)}$ are possible for graphs of polynomial growth with growth
rate (degree) $\delta$, that is, if we assume that every ball of radius $r$
contains only $O(r^\delta)$ vertices. The algorithms use the technique of
low-treewidth pattern covering, introduced by Fomin et al. [FOCS 2016] for
planar graphs; here we show how this strategy can be made to work for graphs
with polynomial growth.
  Formally, we prove that, given a graph $G$ of polynomial growth with growth
rate $\delta$ and an integer $k$, one can in randomized polynomial time find a
subset $A \subseteq V(G)$ such that on one hand the treewidth of $G[A]$ is
$O(k^{1-\frac{1}{1+\delta}} \log k)$, and on the other hand for every set $X
\subseteq V(G)$ of size at most $k$, the probability that $X \subseteq A$ is
$2^{-O(k^{1-\frac{1}{1+\delta}} \log^2 k)}$. Together with standard dynamic
programming techniques on graphs of bounded treewidth, this statement gives
subexponential parameterized algorithms for a number of subgraph search
problems, such as Long Path or Steiner Tree, in graphs of polynomial growth.
  We complement the algorithm with an almost tight lower bound for Long Path:
unless the Exponential Time Hypothesis fails, no parameterized algorithm with
running time $2^{k^{1-\frac{1}{\delta}-\varepsilon}}n^{O(1)}$ is possible for
any $\varepsilon > 0$ and an integer $\delta \geq 3$.
