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?