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

Background

Background

Background

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.

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

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

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

Definitions and Basic properties

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

Key Propositions

Main Results

Idea of proof:

Main Results

Choose x=e

Main Results

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

Main Results

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?

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