The Ground Axiom ($GA$) is the assertion that the universe of
sets ($V$) is not a forcing extension of any inner model $W$ by nontrivial forcing
$P\in W$.


Is $GA$ consistent with any possible behavior of continuum function $\kappa\mapsto 2^{\kappa}$?


It seems in models of $GA$ like $L$ and some other canonical models the growth speed of continuum function is too low (e.g. $L\models GCH$). So the natural question is: 


What is the consistency situation for faster growth speeds of $\kappa\mapsto 2^{\kappa}$? 



Ground Axiom and behaviors of continuum function
In the paper [The ground axiom is consistent with $V\ne{\rm HOD}$](http://dx.doi.org/10.1090/S0002-9939-08-09285-X) (J. D. Hamkins, J. Reitz, W.H. Woodin, PAMS 136(8):2008), we prove that the ground axiom is consistent with $V\neq\text{HOD}$, and remark there that:



> 
> 
> > 
> > The proof of Theorem 1 is
> >  flexible and generalizes in a variety of ways. For example,
> >  not much was used about the specific iteration $\mathbb{P}$. For
> >  establishing GA, we needed to know only that the stage
> >  $\gamma$ forcing was ${\lt}\gamma$-closed, and we could easily
> >  have accommodated $\text{Add}(\gamma,\gamma^{++})$ or
> >  occasionally $\text{Coll}(\gamma,\gamma^{+})$, for example,
> >  without any difficulty in the argument. Also, we needn't
> >  have forced specifically at every regular cardinal stage
> >  $\gamma$, but could have forced at regular cardinals in
> >  some other unbounded pattern. Thus, the argument
> >  establishes that after forcing over $L$ with any of the
> >  usual reverse Easton iterations of closed forcing, one
> >  obtains the Ground Axiom in the extension.
> > 
> > 
> > 
> 
> 
> 


Thus, with this kind of forcing, one can achieve a wide spectrum of patterns in the continuum function. Although the usual Easton forcing itself is a product, rather than an iteration, and will therefore definitely not result in the ground axiom, nevertheless one may transform the usual Easton product into an iteration, by taking big chunks of the forcing at a time. That is, given an Easton function $E$, one looks at sufficient closure points of $E$, and performs the iteration, which performs the usual Easton product between these closure points. The result will be a model of GA with the desired Easton function as the continuum function. I don't know if the details of this argument, however, have ever been written down, and if you were inclined in that direction, I would encourage you to do it (feel free to contact me).


