What is meant by a large subgroup? 
What is the meaning of a large subgroup in following paper?



Is it equal to "maximal subgroup"?

 A: The paper is Hatcher, Allen(1-CRNL); Lochak, Pierre(F-ENS-MI); Schneps, Leila(F-FRANS-LM)
On the Teichmüller tower of mapping class groups. (English summary)
J. Reine Angew. Math. 521 (2000), 1–24.
The subgroup $\Lambda$ they construct is defined before Theorem A on page 3. The meaning of "large" is explained in Theorems B, C, D. It is just the colloquial "large" which has nothing to do with the standard definitions of largeness in the literature.
A: 
Definition: A large group is a group with a finite index subgroup that maps onto the free group $F_2$ of rank $2$.

This is a definition I work with in my research. I'm not sure whether it's the one intended in the paper you cite though.
Edit: I see that you have described something like this definition in your comment.
A: This terminology is not completely standard but:

A subgroup is said to be large if any subgroup that intersects it trivially must be the trivial subgroup.

Source:  https://groupprops.subwiki.org/wiki/Large_subgroup
As I said, the terminology is not standard, so you may find other definitions. For instance, in this paper a subgroup is said to be large if it contains its centralizer.
