Title: The topology on Berkovich affine lines over complete valuation rings,
Abstract: In this article, we give a full description of the topology of the one
dimensional affine analytic space $\mathbb{A}_R^1$ over a complete valuation
ring $R$ (i.e. a valuation ring with "real valued valuation" which is complete
under the induced metric), when its field of fractions $K$ is algebraically
closed. In particular, we show that $\mathbb{A}_R^1$ is both connected and
locally path connected. Furthermore, $\mathbb{A}_R^1$ is the completion of
$K\times (1,\infty)$ under a canonical uniform structure. As an application, we
describe the Berkovich spectrum $\mathfrak{M}(\mathbb{Z}_p[G])$ of the Banach
group ring $\mathbb{Z}_p[G]$ of a cyclic $p$-group $G$ over the ring
$\mathbb{Z}_p$ of $p$-adic integers.