  Self-similar groups provide a rich source of groups with interesting
properties; e.g., infinite torsion groups (Burnside groups) and groups with an
intermediate word growth. Various self-similar groups can be described by a
recursive (possibly infinite) presentation, a so-called finite
$L$-presentation. Finite $L$-presentations allow numerous algorithms for
finitely presented groups to be generalized to this special class of recursive
presentations. We give an overview of the algorithms for finitely $L$-presented
groups. As applications, we demonstrate how their implementation in a computer
algebra system allows us to study explicit examples of self-similar groups
including the Fabrykowski-Gupta groups. Our experiments yield detailed insight
into the structure of these groups.
