Triangle Joins of SP1 Equivalence Relations and their Cost
  Free 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. 
Teodor Stefan Bildea