Arithmetic groups

 The family of  high rank arithmetic groups is a class of groups  playing  an important role in various areas of mathematics.  It  includes  SL(n,Z), for n>2  , SL(n, Z[1/p] ) for n>1, their finite index subgroups and many more.  

   A number of remarkable results about them have been proven including;  Weil local rigidity, Mostow strong rigidity, Margulis  Super rigidity and the Schwartz-Eskin-Farb Quasi-isometric rigidity. 

      We will add  a new type of rigidity  : "first order rigidity".   Namely  if D   is such a non-uniform characteristic zero  arithmetic group and L  a finitely generated group which is elementary equivalent to D  then L is isomorphic to D. 

      This stands in contrast with Zlil Sela's  remarkable work which implies that the free groups, surface groups and hyperbolic groups  ( many of which  are low-rank arithmetic groups) have many

 non isomorphic  finitely generated groups which are elementary equivalent to them. 

     Based on a joint paper with Nir Avni and Chen Meiri ( Invent. Math.217(2019) 219-240).

 

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