A Riesz-Thorin Approach to the Rapid Decay Property for Free Groups

Abstract: We establish $L^p$ bounds for operators associated with the quasi-regular representation of the free group on its Gromov boundary. The $p=2$ case recovers Haagerup's inequality, yielding a new interpolation-theoretic proof of the the Rapid Decay property for the free group.

• The paper introduces an operator-theoretic framework leveraging the Riesz-Thorin interpolation theorem to prove the rapid decay property for free groups.
• It establishes uniform L^p bounds for convolution operators on the Cayley tree boundary, recovering Haagerup's inequality for the L^2 case.
• The approach avoids direct L^2 methods, offering potential extensions to other groups and simplifying analyses in operator algebras.

Riesz-Thorin Interpolation and the Rapid Decay Property in Free Groups

Formal Framework and Background

The paper "A Riesz-Thorin Approach to the Rapid Decay Property for Free Groups" (2606.19880) presents an operator-theoretic proof of the rapid decay (RD) property for free groups, leveraging the Riesz-Thorin interpolation theorem to establish $L^p$ bounds for operators associated with quasi-regular representations on the Gromov boundary of the Cayley tree.

The free group $I$ of rank $r$ acts transitively on the homogeneous tree $T$ of degree $2r$, and its Gromov boundary $\partial T$ is constructed as the set of right-infinite reduced words. The boundary admits a natural compact topology and a quasi-invariant Borel probability measure $v$ with respect to the action of the group.

The main objects of study are the quasi-regular representation $T: I \to U(L^2(\partial T, v))$ and covariant representations with respect to the action on $C(\partial T)$. The paper avoids the direct use of $L^2$-specific arguments, working instead within $I$0 and $I$1 regimes by construction of suitable partial representation operators $I$2 and their tensor products.

Operator Bounds and the Riesz-Thorin Theorem

The RD property for the free group is cast as an operator-norm estimate on convolution operators with support in the sphere $I$3:

$I$4

where $I$5 is the group convolution operator associated with function $I$6 supported in $I$7.

The central innovation is in decomposing $I$8 as $I$9 and constructing tensor product operators $r$0. The argument proceeds by establishing uniform $r$1 bounds for $r$2:

$r$3

for all $r$4. The $r$5 and $r$6 cases are verified directly using geometric considerations of the tree and boundary, exploiting the cocycle identity and structure of the sets $r$7 and $r$8. The uniqueness property identified in Lemma 3.2 ensures non-redundancy in operator support, critical for norm estimates.

The Riesz-Thorin interpolation theorem is then applied to interpolate between the $r$9 and $T$0 bounds, yielding tight uniform $T$1 estimates, and thus establishing the RD property for $T$2 with no dependence on deep $T$3-harmonic analysis, but rather through interpolation theory.

Numerical and Structural Results

The paper concretely proves that $T$4 for all $T$5, and the $T$6 case fully recovers Haagerup's inequality. This approach offers an alternative to classical proofs of RD property, showing that operator norm bounds can be derived without direct recourse to $T$7 inner product structure or Plancherel identity, but purely via functional-analytic interpolation and the combinatorics of tree boundaries.

No contradictory claims are presented; the paper asserts the scope of uniform LP-bounds within the framework provided. The results are structurally robust, leveraging the geometry and topology of the boundary, and demonstrating full compatibility with previous literature on ergodic and representation-theoretic properties of free groups.

Implications and Potential Extensions

This work has several implications. First, the operator-theoretic proof framework via interpolation opens the possibility of approaching RD property proofs for more general classes of groups, particularly hyperbolic groups with similar boundary representations. The author explicitly notes the potential for extending this method to boundary representations of hyperbolic groups, where existing proofs are often more intricate and less amenable to interpolation-based approaches.

On the practical side, the interpolation-theoretic framework may facilitate computational bounds in the study of C*-algebras associated with free groups and their representations. Given the centrality of the RD property in property (RD) and its links to the metric approximation property in noncommutative geometry, this result provides a streamlined avenue for operator-norm estimation in group C*-algebras.

Theoretically, this advances the understanding of boundary representations, quasi-regular representations, and their LP norm behavior. It refines the analytic toolkit available for studying group actions on boundaries, aiming at further generalizations in geometric group theory and noncommutative analysis.

Future developments may include further optimization of operator-norm bounds for groups beyond free groups, exploration of different interpolation bases in noncommutative settings, and constructive applications in spectral theory, group C*-algebras, and harmonic analysis on trees and boundaries.

Conclusion

The paper delivers a rigorous operator-theoretic proof of the rapid decay property for free groups via the Riesz-Thorin interpolation theorem, providing uniform LP bounds and recovering classical Haagerup inequalities. This approach distinguishes itself by avoiding direct $T$8 arguments, instead leveraging functional-analytic methods and boundary geometry. The technique invites further exploration in broader classes of groups and boundary representations, offering both theoretical and practical advances in group representation theory and operator algebras.