M.Sc Thesis
M.Sc Student Nachum Ido Coarse Symmetries of Groups Department of Mathematics Professor Uri Bader

Abstract

Given a group G, we suggest an alternative approach for defining the groups Aut(G), Comm(G) and Q.I.(G). We do this through the use of geometrical couplings of the group G with itself, an idea coming from Gromov's quasi isometry criterion for groups: given two finitely generated groups , they are quasi isometric if and only if there exist commuting, cocompact, proper and continuous actions of them on some locally compact Hausdorff space. The action is a topological coupling. Following these ideas, we finally define a new group M.E.(G) on the set of all measure equivalences of  G with itself: Given two discrete groups, we say that they are   measure equivalent if there exists commuting, essentially free measure preserving actions of them with finite fundamental domains on some infinite Lebesgue space. The action is a measured coupling._____________________________________________________________________________