  The distinguishing number $\operatorname D(G)$ of a graph $G$ is the least
cardinal $d$ such that $G$ has a labeling with $d$ labels which is only
preserved by the trivial automorphism. We show that the distinguishing number
of infinite, locally finite, connected graphs $G$ with infinite motion and
growth $o \left(\frac{n^2}{\log_2 n} \right)$ is either $1$ or $2$, which
proves the Infinite Motion Conjecture of Tom Tucker for this type of graphs.
The same holds true for graphs with countably many ends that do not grow too
fast. We also show that graphs $G$ of arbitrary cardinality are
$2$-distinguishable if every nontrivial automorphism moves at least uncountably
many vertices $m(G)$, where $m(G) \geq
\left\vert\operatorname{Aut}(G)\right\vert$. This extends a result of Imrich et
al. to graphs with automorphism groups of arbitrary cardinality.
