[Paper Review] N-step energy of maps and fixed-point property of random groups

Hiroyasu Izeki, Takefumi Kondo+1 more|arXiv (Cornell University)|Oct 22, 2012

Geometric and Algebraic Topology14 references4 citations

TL;DR

This paper establishes that random groups constructed via the graph model using expanders possess the fixed-point property for a broad class of CAT(0) spaces, including all Euclidean buildings of uniformly bounded dimension. By refining Gromov’s criterion on n-step energy growth and estimating the Wang invariant (δ) of tangent cones in these buildings, the authors prove that such groups fix global points in all CAT(0) spaces with δ < 1, extending previous results on Hilbert spaces and property (T).

ABSTRACT

We prove that a random group of the graph model associated with a sequence of expanders has fixed-point property for a certain class of CAT(0) spaces. We use Gromov's criterion for fixed-point property in terms of the growth of n-step energy of equivariant maps from a finitely generated group into a CAT(0) space, to which we give a detailed proof. We estimate a relevant geometric invariant of the tangent cones of the Euclidean buildings associated with the groups PGL(m,Q_r), and deduce from the general result above that the same random group has fixed-point property for all of these Euclidean buildings with m bounded from above.

Motivation & Objective

• To establish the fixed-point property for random groups in the graph model across a broad class of CAT(0) spaces beyond Hilbert spaces.
• To generalize Silberman’s proof technique for property (T) by replacing large deviation bounds with the central limit theorem, enabling treatment of graphs with degree-two vertices.
• To extend spectral gap inequalities from Hilbert space targets to general CAT(0) space targets, broadening applicability.
• To compute or upper-bound the geometric invariant δ (Wang invariant) of tangent cones in Euclidean buildings associated with PGL(m, Q_r), showing it is strictly less than 1 for bounded m.
• To demonstrate that the fixed-point property for these buildings follows from the general criterion on n-step energy growth and the boundedness of δ.

Proposed method

• Adapts Gromov’s criterion for fixed-point property, which links the growth of n-step energy of equivariant maps into a CAT(0) space to the existence of a global fixed point.
• Replaces Silberman’s large deviation inequality for Bernoulli walks with the central limit theorem to estimate the probability that random walks on graphs remain close to the origin.
• Introduces a probabilistic framework using weighted measures P_G^n(l) on graph distances, derived from random walks, to bound the energy of equivariant maps.
• Estimates the n-step energy E_μ(f) of equivariant maps using spectral gap λ_1(G,Y) of the graph G with respect to the target space Y.
• Defines and computes the radial distortion of metric cones as a proxy for the Wang invariant δ, which controls the δ-invariant of tangent cones.
• Applies the general fixed-point criterion to Euclidean buildings associated with PGL(m, Q_r), showing δ(TC_pY) < 1 uniformly for all p and bounded m.

Experimental results

Research questions

• RQ1 Do random groups in the graph model have the fixed-point property for CAT(0) spaces beyond Hilbert spaces?
• RQ2 Can Gromov’s n-step energy criterion be effectively applied to prove fixed-point properties for random groups acting on non-Hilbert CAT(0) spaces?
• RQ3 What is the value of the Wang invariant δ for tangent cones of Euclidean buildings associated with PGL(m, Q_r) for prime r and bounded m?
• RQ4 Does the fixed-point property for Euclidean buildings follow from the boundedness of δ and the n-step energy growth criterion?
• RQ5 Can the probabilistic argument used for Hilbert spaces be adapted to CAT(0) targets by replacing large deviation bounds with central limit theorems?

Key findings

• The n-step energy of equivariant maps into a CAT(0) space grows sublinearly under the fixed-point criterion, implying the existence of a global fixed point.
• For random groups built from expanders via the graph model, the n-step energy growth condition is satisfied uniformly for all CAT(0) spaces with δ ≤ δ₀ < 1.
• The Wang invariant δ of the tangent cones of Euclidean buildings associated with PGL(m, Q_r) is bounded above by a constant strictly less than 1, uniformly for all m bounded above.
• The fixed-point property holds for all Euclidean buildings of dimension bounded from above, due to the uniform upper bound on δ.
• The probabilistic argument is simplified by replacing large deviation inequalities with the central limit theorem, allowing graphs with degree-two vertices.
• The result extends beyond Hilbert spaces and property (T), showing fixed-point property for a strictly larger class of CAT(0) spaces, including non-symmetric and non-locally compact ones.