Distinguishing graphs with infinite motion and nonlinear growth

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.