Bouquet of Two Circles and Based covers

Question

I need help with the following: Let $B$ denote the bouquet of two circles labeled by $a,b$. We regard $\pi_{1}(B)\cong \langle a,b | -\rangle $ where the basepoint of $B$ is its vertex, and the two edges determine paths correspoinding to basis elements $a,b$ of the free group. Sketch the based cover $\hat{B}\rightarrow B$ such that $\pi_{1}(\hat{B})$ corresponds to the subgroup

1. $\langle aaabbb,aaaabbbb\rangle$
2. $\langle bbbbb, bababab\rangle$

Any hints?

Answer 1

Hint: find a smallest labeled graph such that $a^3b^3$ and $a^4b^4$ represent nontrivial loops. Expand it to a cover without adding any more loops (try to mimic the way the universal cover of $B$ avoids adding any new loops to the single vertex, no edges graph $\{*\}$). If you don't know what the universal cover of $B$ looks like, find it first.