The Details:
Definition 1: A class $\mathcal{G}$ of groups satisfies the Tits alternative if for any $G$ in $\mathcal{G}$ either $G$ has a free, non-abelian subgroup or $G$ has a solvable subgroup of finite index.
Example 1: The class of finitely generated linear groups satisfies the Tits alternative.
For my PhD, I plan to establish whether or not the Tits alternative holds for a certain class, $\mathcal{G}_0$ say, of cyclically presented groups. I'm new to this, however, since I'm only a few months into the PhD and I've had to change topics as, already, all of my previous topic has been subsumed by some yet-to-be-published work (as of $6^{\text{th}}$ Feb, $2018$) by someone else.
The Question(s):
What is the big picture for the Tits alternative is in terms of Combinatorial Group Theory (or wider)? What is the motivation for studying it?
Thoughts:
Classes of groups similar to $\mathcal{G}_0$ have been studied in terms of the Tits alternative before and, in some cases, it has been fully established, according to my supervisor, who said that the Tits alternative is quite an important property.
My guess is that what's going on is something similar to how exact sequences, by means of extensions, motivate the study & classification of solvable groups.
Please help :)