Metric criteria for fixed price of countable groups

TL;DR

The paper develops a general framework to establish fixed price $1$ for countable groups by linking left-invariant metrics, limit-amenability, and cost via Poisson suspensions. It extends weak containment to infinite-measure actions and uses Horofunctions and a shift-space formulation to build actions that are limit-amenable and doubly recurrent, then connects these properties to normalized cost and max-cost bounds. A central criterion shows that if a group admits a limit-amenable, partially doubly recurrent action with normalized cost $p$, then its max-cost does not exceed $p$, yielding fixed price $1$ in the case $p=1$; this is applied to metric groups and product groups under mild growth hypotheses. The results unify and extend prior Poisson-suspension methods and provide a versatile route toward the fixed-price problem for broad classes of groups, including all direct products $\Gamma\times\Gamma$ and many non-amenable groups with suitable metrics or growth conditions.

Abstract

We establish general criteria for a countable group $Γ$ to have fixed price 1 depending on a choice of left-invariant proper metric on $Γ$. We apply this criterion to show that if $Γ_1,Γ_2$ are two countable groups satisfying a certain growth condition then $Γ_1\times Γ_2$ has fixed price 1. For example, $Γ\times Γ$ has fixed price 1 for any countable group $Γ$.

Theorems & Definitions (194)

• Corollary 1.1
• Remark 1
• Definition 1
• Theorem 1.2
• Remark 2
• Remark 3
• Theorem 1.3
• Definition 2
• Definition 3
• Remark 4