Topological properties of strongly topological gyrogroups and geometric properties of strongly generated gyrogroups

Abstract

In this dissertation, we study some topological properties of strongly topological gyrogroups and some geometric properties of strongly generated gyrogroups. First, we embed a strongly topological gyrogroup into a path-connected and locally path-connected strongly topological gyrogroup as a closed subgyrogroup and study some consequences.

Next, we extend the notion of word metric, which has been defined for groups, to the class of strongly generated gyrogroups. This leads to the gyrogroups' version of the fundamental lemma of geometric group theory. Lastly, we show that every locally compact Hausdorff strongly topological gyrogroup admits a unique Haar measure. Then we study the convolution operation on the space of Haar integrable functions. It turns out that, when equipped with the convolution operation, the space of Haar integrable functions becomes a nonassociative normed-algebra.