The purpose of this paper is to construct hyperbolic integer homology 3-spheres where the injectivity radius is arbitrarily large for nearly all points of the manifolds. The key claim is to construct a sequence \(\{N_{n}\}\) of manifolds such that for all large \( n,\) the manifold \(N_{n}\) has a complete hyperbolic metric of finite volume and moreover \[ {\lim} _{n \rightarrow \infty}{\text{vol}(thin_{R} N_{n}) \over {\text{vol}(N_{n})}}=0. \] This leads to the main result: Given positive constants \(R\) and \(\epsilon\) and a finitely-generated abelian group \(A\), there exists a closed hyperbolic 3-manifold \(M\) where \(H_{1}(M;{\mathbf {Z}})=A\) and \[ {{\text{vol}}(thin_{R}M) \over {\text{vol(M)}}} < \epsilon. \] The result is closely related to a conjecture of Bergeron and Venkatesh on growth of torsion in the homology of arithmetic hyperbolic 3-manifolds. As a consequence of the main result, the authors show that there exist closed hyperbolic 3-manifolds \(M_{n}\) which Benjamini-Schramm converge to \({\mathbf H}^{3}\) where \(\tau(M_{n})/\text{vol}(M_{n}) \rightarrow 0\) as \(n \rightarrow \infty\) and in particular the limit is not \(\tau^{(2)}({\mathbf H}^{3})=1/{6\pi}.\) Here \(\tau(M)\) is the logarithm of the analytic torsion and \(\tau^{(2)}({\mathbf H}^{3})\) is the \(L^{2}\)-analytic torsion of \({\mathbf H}^{3}\). At the end of this paper they give results by computational experiments which strongly support the conjecture as well as certain generalizations to nonarithmetic manifolds.