Groups defined by automata

Finite automata have been used effectively in recent years to define infinite groups. The twomain lines of research have as their most representative objects the class of automatic groups (in-cluding word-hyperbolic groups as a particular case) and automata groups (singled out among themore general self-similar groups).The first approach implements in the language of automata some tight constraints on the ge-ometry of the group’s Cayley graph, building strange, beautiful bridges between far-off domains.Automata are used to define a normal form for group elements, and to monitor the fundamentalgroup operations.The second approach features groups acting in a finitely constrained manner on a regular rootedtree. Automata define sequential permutations of the tree, and represent the group elements them-selves. The choice of particular classes of automata has often provided groups with exotic behaviourwhich have revolutioned our perception of infinite finitely generated groups.