A vertex in a graph is called central if it minimizes its maximum distance to the other vertices. The radius of a graph $G$ is the largest distance between a central vertex and the other vertices, and it is denoted by $rad(G)$. In the center problem, we are asked to find a central vertex. We study the fine-grained complexity of the center problem on graphs with small Gromov hyperbolicity. Roughly, the Gromov hyperbolicity of a graph represents how close, locally, it is to a tree, from a metric point of view. It has applications in the design of approximation algorithms. In particular, there is a linear-time algorithm that for every $δ$-hyperbolic graph $G$ outputs some vertex at distance at most $rad(G) + 5δ$ to the other vertices [Chepoi et al, SoCG'08]. However, a linear-time algorithm for computing a central vertex is known only for $0$-hyperbolic graphs, whereas its existence was ruled out for $2$-hyperbolic graphs under the Hitting Set Conjecture of [Abboud et al, SODA'16]. Our main contribution in the paper is a linear-time algorithm for computing a central vertex in the class of $\frac 1 2$-hyperbolic graphs. Furthermore, we rule out the existence of such an algorithm for $1$-hyperbolic graphs, under the Hitting Set Conjecture, thus completely settling all the cases left open.