What is known about maximal free subgroups of surface groups? Let $\Gamma_g=< a_1,...,a_g,b_1,...,b_g | \prod_{i=1}^g [a_i,b_i]>$ (a surface group). What is known about maximal free subgroups of $\Gamma_g$  for $g>1.$ (I.e. free subgroups which are not properly embedded into any other free subgroup)?
For example,


*

*Is $<a_1,...,a_g,b_1,...,b_{g-1},b_g^2>$ free? Maximal?

*What are the lower, upper bounds on ranks of the maximal free subgroups?

*Can the free subgroups of the minimal rank be classified somehow?

 A: *

*The subgroup is free, and therefore not maximal (see 2.). To see this, the subgroup generated by all the generators except $b_g$ is the fundamental group of a subsurface obtained by cutting along a curve (dual to $b_g$). This surface lifts to a 2-fold cover dual to the curve, and the subgroup is obtained by adding another element corresponding to $b_g^2$, so it has infinite index in the 2-fold cover (since its rank is too small to generate). 

*Maximal rank free subgroups (i.e. subgroups of infinite index) must be infinitely generated. More generally, this holds for lattices in $PSL_2(\mathbb{C})$ (the argument they gives works for $PSL_2(\mathbb{R})$, and is even easier, since finitely-generated subgroups are geometrically finite). 

*Since maximal free subgroups have infinite rank, I don't think there will be a nice description of them. They can't be normal, since the only possible quotient groups could have no proper non-trivial subgroups, so must be finite cyclic groups of prime order, a contradiction.  
