Some examples of irreducible homology 3-spheres that bound  **smooth** contractible 4-manifolds are listed in the comment to problem 4.2 in  [Kirby's problem list](http://math.berkeley.edu/~kirby/problems.ps.gz), and all of them happen to occur among the [Brieskorn spheres](http://www.maths.ed.ac.uk/~aar/papers/milnbries.pdf) $\Sigma(p,q,r)$ modelled on $\widetilde{SL}\_2(\mathbb R)$,
i.e. such that $\frac{1}{p}+\frac{1}{q}+\frac{1}{r}<1$ (except for the standard $S^3$, of course). 


 **Question.** Are there any known homology 3-spheres that bound smooth contractible 4-manifolds and are modelled on other geometries, e.g. NIL or the hyperbolic space?


EDIT: Glazner in the paper "Uncountably many contractible 4-manifolds" constructed some other examples but I cannot recognize the geometry. (Glazner's six page paper
is easily googlable by title, and it gives an explicit representation for the fundamental group, denoted $G\_n$ on page 40). 



Irreducible homology 3-spheres that bound smooth contractible manifolds
Yes, there's loads of other contractible 4-manifolds bounding homology spheres with other geometries.


For example $1$-surgery on the Stevedore knot is a hyperbolic manifold with volume $1.3985\cdots$. This bounds a contractible manifold by the same kind of arguments Casson and Harer used to cook up their big list of Mazur manifolds. There are other examples like this that occur in my arXiv preprint: <https://arxiv.org/abs/0810.2346> Some are geometric, some have incompressible tori.


I don't think any Nil manifolds bound contractible 4-manifolds. Crisp and Hillman determined all the Nil manifolds that embed smoothly (or topologically) in the $4$-sphere. The only Nil manifolds on that list have non-trivial homology. Crisp & Hillman "Embedding Seifert fibred 3-manifolds and Sol-manifolds in 4-space".


