  Using topological notions of translation-like actions introduced by
Schneider, we give a positive answer to a geometric version of Burnside problem
for locally compact group. The main theorem states that a locally compact group
is non-compact if and only if it admits a translation-like action by the group
of integers $\mathbf Z$. We then characterize the existence of cocompact
translation-like actions of $\mathbf Z$ or non-abelian free groups on a large
class of locally compact groups, improving on Schneider's results and
generalising Seward's.
