The context & motivation for the Tits alternative in combinatorial group theory 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 :)
 A: The Tits alternative, first proven by Jacques Tits for linear groups, says the following:

If $G$ is a finitely generated subgroup of a linear group then $G$ is either virtually soluble* or contains a non-abelian free subgroup.

In geometric and combinatorial group theory, "being virtually-$\mathcal{P}$" for some property $\mathcal{P}$ is basically the same as "being $\mathcal{P}$". Now, a group cannot both have a non-abelian free subgroup and be virtually-soluble (why?). Therefore, the Tits alternative says that containing a free subgroup is the only obstruction to solubility which a finitely generated subgroup of a linear group can have.
The Tits alternative is considered a "nice", linear-like property. Therefore, showing that a class of groups satisfies the Tits alternative shows that you are dealing with a nice class of groups (and cyclically presented groups are weird! If they were also nice then this would be...nice.).
Why do we want to show linear-like properties? Well, as an example, it is a famous open question of Gromov whether or not hyperbolic groups are residually finite**. One way of showing this would be to prove that they are all linear (as linear groups are residually finite). So a first step towards proving linearity would be to prove that they satisfy the Tits alternative. And they do. Yay! On the other hand, Misha Kapovich proved that there are non-linear hyperbolic groups. So hyperbolic groups sit in a curious middle ground, along with $\operatorname{Out(F_n)}$ and mapping class groups... To add some fuel to the fire - Agol, Wise and others proved that masses of hyperbolic groups are indeed linear (basically, every hyperbolic group you've ever heard of apart from those synthetically cooked up by Kapovich to be non-linear). So hyperbolic groups are "mostly" linear, and are morally linear. Except they some are not linear. This is an (IMO, exteremly) interesting state of affairs.
*In American English, solvable.
**You may of course ask "residual finiteness: why do we care?"...
