Rigidity of amalgamated free products in measure equivalence

TL;DR

This paper introduces a new class of ME rigid groups formed by amalgamating groups with measure equivalence rigidity, and investigates their measure equivalence properties.

Contribution

It constructs ME rigid groups via amalgamated free products and analyzes measure equivalence for these groups, expanding understanding of rigidity in this context.

Findings

Constructed new ME rigid groups through amalgamation.

Identified measure equivalence classes related to these groups.

Extended rigidity concepts to amalgamated free products.

Abstract

A discrete countable group \Gamma is said to be ME rigid if any discrete countable group that is measure equivalent to \Gamma is virtually isomorphic to \Gamma. In this paper, we construct ME rigid groups by amalgamating two groups satisfying rigidity in a sense of measure equivalence. A class of amalgamated free products is introduced, and discrete countable groups which are measure equivalent to a group in that class are investigated.