# Colloquium abstracts

**Kingshook Biswas***

ISI Kolkata

October 28, 2021

**Quasi-metric antipodal spaces and maximal Gromov hyperbolic spaces**: A quasi-metric antipodal space $(Z, ho_0)$ is a compact space $Z$ with a continuous quasi-metric $ho_0$ which is of diameter one, and which is antipodal, i.e. for any $xi in Z$ there exists $eta in Z$ such that $ho_0(xi, eta) = 1$. The quasi-metric $ho_0$ defines a positive cross-ratio function on the space of quadruples of distinct points in $Z$, and a homeomorphism between quasi-metric antipodal spaces is said to be Moebius if it preserves cross-ratios. A proper, geodesically complete Gromov hyperbolic space $X$ is said to be boundary continuous if the Gromov inner product extends continuously to the boundary. Then the boundary $partial X$ equipped with a visual quasi-metric is a quasi-metric antipodal space. The space $X$ is said to be maximal if for any proper, geodesically complete, boundary continuous Gromov hyperbolic space $Y$, if there is a Moebius homeomorphism $f :partial Y o partial X$, then $f$ extends to an isometric embedding $F : Y o X$. We call such spaces maxim