1. On Property L On Property L School of Mathematics School of Mathematics Fudan University Fudan University Xiaoman Chen & Xianjin Wan

2. Background

3. Background

4. Background

5. Background What is property L? The property L is a local estimation property for us to estimate the norm of any operator in Roe algebra. The property L is a local estimation property for us to estimate the norm of any operator in Roe algebra.

6. Background The Box space : Let Γ be a finite generated residually finite group

7. Background Question: Is this mapping can be extended to the reduced Roe Algebras? In Gong-Wang-Yu’s paper “Geometrization of the Strong Novikov Conjecture of Residually finite groups”, they proved that

8. Definitions and Basic properties It is not difficult to prove that if Γ has finite asymptotic dimension, then the above lifting can be extended to the reduced Roe algebra. Generalize the finite asymptotic case, Guoliang Yu introduced the following definition

9. Definitions and Basic properties

18. Problems 1. What kinds of finite generated groups are being of property L? 2. Do the operations of groups preserve property L?

19. Key Propositions

28. Main Results

29. Idea of proof:

30. Main Results

31. Choose x=e

32. Main Results

33. Using the above Proposition and the infinite union theorem, we have

34. Main Results

35. Further Problems 1. Let Γ be a finite generated residually finite group with the property L. Are the reduced Roe algebra and maximal Roe algebra of its box space same? 2. Can we prove the Coarse Baum-Connes Conjecture in the case of the property L?

36. Thank you ! Thank you !