2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/74492Free joins of measure preserving equivalence relations, with or without amalgamation were introduced by Gaboriau, who studied their cost and obtained a formula for the cost of free products of groups. We introduce the notion of triangle join of measure preserving equivalence relations and study their cost. As a consequence we show that the cost of Thompson's group G is 1.Added computation of the cost of Thompsons group GGroup TheoryDynamical Systems37A20,20E06,20F67text