Products of Infinite Countable Groups Have Fixed Price One

TL;DR

The paper resolves Gaboriau’s fixed price problem for the product of two infinite countable groups by constructing a Poisson horoball process as a weak limit of factors of i.i.d. The propagation method yields a low-cost graphing inside horoballs, and a small percolation fuses these components through the infinite touching property, achieving a connected graphing with cost arbitrarily close to $1$. It extends the induction principle to random multisets (with multiple points) and introduces perturbed diamonds to circumvent growth assumptions, ensuring convergence to horoballs of type II. The approach provides a self-contained, direct proof avoiding amenability or double-recurrence, and establishes fixed price $1$ for all infinite countable group products, with connections to prior IPVT-based results in special settings. This advances the understanding of cost in measured group theory and offers robust tools for analyzing weak factors of i.i.d. in geometric group constructs.

Abstract

We prove that the product of any two infinite countable groups has fixed price one. This resolves a problem posed by Gaboriau. The proof uses the propagation method to construct a Poisson horoball process as a weak limit of a sequence of factors of i.i.d. An essential tool for avoiding any growth conditions is the convergence in the sense of point processes of closed pointed subsets, which is a notion from stochastic geometry. Then, a graphing of the horoball process is constructed with arbitrarily small expected degree, by connecting the points of each horoball first, and then adding a percolation with small intensity. The connectedness of this graphing is ensured by proving that the resulting horoballs have the infinite touching property almost surely, if the metric and the other parameters of the construction are chosen carefully. Direct simple proofs are given that do not rely on sophisticated results like amenability and double-recurrence, which are used in related works. Also, to manage the overlapping of the horoballs, a generalization of the induction lemma is presented for random multisets of a group.

Figures (4)

• Figure 1: In the Poincaré model of the hyperbolic plane, a Boolean model containing balls of fixed radius is shown on the left, and a Poisson horoball process is shown on the right. The centers of the balls are shown in red. See \ref{['intro:poisson']}.
• Figure 2: In the 3-regular tree, a Boolean model containing balls of radius 2 is shown on the left, where the centers of the balls are shown in red. On the right, a Poisson horoball process is shown. See \ref{['intro:poisson']}.
• Figure 3: Horoballs in the plane with the metric $\rho_c$\ref{['eq:rho']}, where $c=\frac{1}{2}$ (see the end of \ref{['subsec:boundary']}). On the left, the horoballs of type I are shown, which are corners in the plane. On the right, the horoballs are of type II, and are half-planes with slope $\pm c$. Two paths with slope $c$ are also depicted, that are in bounded distance from each other.
• Figure 4: The paths $\eta$ and $\eta'$ in $G$ (left) and $G'$ (right) respectively, used in the proof of \ref{['lem:infiniteTouching']}.

Theorems & Definitions (41)

• Theorem 1.1
• Theorem 1.2
• Proposition 1.3: See Ga24
• Proposition 1.5
• proof : Sketch of the proof
• Theorem 1.6: Induction Formula for Marked Point Processes
• Proposition 1.8
• proof : Proof of \ref{['prop:growth']}
• Lemma 2.1
• proof