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Floyd boundary:
Floyd length: We give a length to edges determined by the word distance.
The Floyd metric is then the metric space with the lengths. Using this we define the Floyd boundary.
This is defined for
- finite groups
- products of two infinite groups
- any amenable group
- mapping class groups
We can define something like the Hausdorff dimension for the Floyd boundary.
Random walks
- we place a particle at identity
- walk to another location given by a function
Theorem
A countable group is amenable if the spectral radius calculated by the random walk is equals to 1.
Taking the inverse of the spectral radius give something called the r-Green function.
Branching Random walks
We allow particles to split into smaller ones. They split into an average of r particles. I feel like this is useful, because the amount of particle grows exponentially.
We define the trace of a random walk. This can be a true subset of the Cayley Graph.
Relatively Hyperbolic group
A finitely generated group is relatively hyperboic if it acts properly on a proper hyperbolic space X and some conditions are fulfilled.
References
- Floyd 1980
- karlson