In geometric/combinatorial group theory, Grigorchuk's 2-group is a fundamental example. It originally was constructed as a particularly elegant infinite finitely generated torsion group. Then Grigorchuk showed it had intermediate growth, answering Milnor's problem. For a time it was seen as the universal counterexample in group theory. But now it has spawned a theory of groups acting on rooted trees, self-similar groups and branch groups. Pierre de la Harper has an entire chapter of his book devoted to this group.

In geometric/combinatorial group theory, Grigorchuk's 2-group is a fundamental example. It originally was constructed as a particularly elegant infinite finitely generated torsion group. Then Grigorchuk showed it had intermediate growth, answering Milnor's problem. For a time it was seen as the universal counterexample in group theory. But now it has spawned a theory of groups acting on rooted trees, self-similar groups and branch groups. Pierre de la Harpe has an entire chapter of his book devoted to this group.