Ping-pong lemma

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Statement

Statement in terms of two elements and free groups

Suppose G is a group with a group action on a set X. Further, suppose A and B are two subsets of X such that neither is contained in the other. Further, suppose a,b \in G are elements such that:

  • b^n(A) \subseteq B \ \forall \ n \ne 0.
  • a^n(B) \subseteq A \ \forall \ n \ne 0.

Then, the subgroup \langle a, b \rangle is a free group with \{a,b\} as a freely generating set.

Statement in terms of two subgroups and free products

Suppose G is a group with a group action on a set X. Further, suppose A and B are two subsets of X such that neither is contained in the other. Further, suppose H, K \le G are subgroups such that:

  • g(A) \subseteq B \ \forall \ g \in K.
  • g(B) \subseteq A \ \forall \ g \in H.

Then, the subgroup \langle H, K \rangle is an internal free product of the subgroups H and K.

Related facts

Applications