The Divergence of the Special Linear Group over a Function Ring

Abstract

We compute the divergence of the finitely generated group SL _n (𝒪_𝒮), where 𝒮 is a finite set of valuations of a function field and 𝒪_𝒮 is the corresponding ring of 𝒮-integer points. As an application, we deduce that all its asymptotic cones are without cut-points.

Key Words:

• Asymptotic cones
• Divergence
• Function fields
• Special linear group

2010 Mathematics Subject Classification:

• Primary: 20F65
• Secondary: 20F69, 15B33

ACKNOWLEDGMENTS

I am very grateful to Yves de Cornulier for various helpful comments and suggestions. I would also like to thank Pierre-Emmanuel Caprace for useful discussions. Part of this work was done during Park City Summer Program on Geometric Group Theory, and its organizers are gratefully acknowledged.

Notes

Communicated by A. Olshanskii.