Abstract: Extremal hyperbolic manifolds and orbifolds, namely those which have the smallest possible volume, were a subject of interest for a long time. A classical example of such an orbifold is the quotient of Klein's quartic by its group of automorphisms, which can be also obtained as G\SO(1,2) where G is so-called Hurwitz group (2,3,7). This orbifold and other known examples of small volume in dimension 3 are all arithmetic by which we mean that they are uniformized by arithmetic subgroups of SO(1,n).

The goal of my recent research can be formulated as to obtain the higher dimensional analogues of the Hurwitz group. As a result, for each even dimension n we prove that there exists a unique minimal arithmetic hyperbolic n-orbifold and give a formula for its generalized Euler characteristic. The argument uses G. Prasad's volume formula for arithmetic quotients of semi-simple groups and Bruhat-Tits theory.

In the talk I will explain main ideas of the proof and also discuss applications and possible generalizations of the results.