TITLE: Statements about groups proved using semigroups
QUESTION [15 upvotes]: Question. Has a statement about groups ever been proved using the theory of semigroups?

By "a proof using the theory of semigroups" I do not mean that some steps in the proof are in fact statements about semigroups rather than groups. What I'm looking for are proofs which involve semigroups in a substantial way, say by constructing some semigroup related to the group of interest.

REPLY [4 votes]: There is a theorem saying that if we take a Zariski-dense subgroup $\Gamma$ in a semisimple real Lie group $G$, for example $G = \operatorname{SL}_d(\mathbb{R})$, the set of so-called "loxodromic" elements in $\Gamma$ is still Zariski-dense in $G$. (In the case of $G = \operatorname{SL}_d(\mathbb{R})$, "loxodromic" means "diagonalizable with eigenvalues of distinct modulus").
In order to prove it, one actually proves the same statement for semigroups. See Proposition 5.11 and Remark 5.14 in Y. Benoist and J.F. Quint's book "Random walks on reductive groups".