  Upper bounds for the probabilities $\mathbb{P}(F\geq \mathbb{E} F + r)$ and
$\mathbb{P}(F\leq \mathbb{E} F - r)$ are proved, where $F$ is a certain
component count associated with a random geometric graph built over a Poisson
point process on $\mathbb{R}^d$. The bounds for the upper tail decay
exponentially, and the lower tail estimates even have a Gaussian decay.
  For the proof of the concentration inequalities, recently developed methods
based on logarithmic Sobolev inequalities are used and enhanced. A particular
advantage of this approach is that the resulting inequalities even apply in
settings where the underlying Poisson process has infinite intensity measure.
