It is well known that for a closed hyperbolic 3-manifold $M$ the rank of $\pi_1(M)$ is bounded above by some universal constant $K$ times the volume of $M$. Using similar methods, i.e. the thick-thin decomposition of $M$, one can also show that the Heegaard genus of $M$ is bounded above by a universal constant times the volume of $M$. (I believe that Thurston showed this first, though I am not sure as to how.)

I am looking to construct a sequence of (closed) hyperbolic 3-manifolds, say {$M_n$} such that the volume grows linearly in the Heegaard genus of $M_n$. That is, I am trying to show that a linear bounded on Heegaard genus in terms of volume is the best one can do. So far, I am having some trouble constructing such an example.

Does anyone have a good method or reference for constructing such an example? Also, another approach to the problem would be wonderful (short of solving the rank versus Heegaard genus conjecture of hyperbolic 3-manifolds, of course).


Agol says: Take a 3-manifold group that maps onto a free group, and take induced covers (the rank and volume will both grow linearly). See this paper of Lackenby.


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