I'm working on the Heegaard splitting of some 3-manifold and I'm currently stucked on the Weeks manifold. It is a closed orientable hyperbolic 3-manifold with the smallest volume, obtained by (5, 2) and (5, 1) Dehn surgeries on the Whitehead link.
I found a presentation of the fundamental group of the Weeks manifold, it is $$\pi_1(M_W)=\langle a,b\mid a^2b^2a^2b^{-1}ab^{-1}=1,a^2b^2a^{-1}ba^{-1}b^2=1\rangle.$$
What is the rank of this group? Do you know any better presentation of the Weeks manifold's fundamental group? Or do you know the better ways to compute the Heegard genus of this manifold?
Rank of this group is clearly $2$: If it were of rank $\le 1$, the group would be cyclic (finite or infinite), which cannot be the fundamental group of a hyperbolic manifold (complete, finite volume). In general, SnapPea can find Heegaard diagrams for you: https://t3m.math.uic.edu/. The Weeks manifold has genus 2, see for instance
Vesnin, A. Yu.; Mednykh, A. D., The Heegaard genus of hyperbolic 3-manifolds of small volume, Sib. Math. J. 37, No. 5, 893-897 (1996); translation from Sib. Mat. Zh. 37, No. 5, 1013-1018 (1996). ZBL0887.57021.