Property (T) above one-third density in random free-product quotients

Abstract

Fix at least three nontrivial finitely generated groups and a finite inverse-closed generating alphabet of nonidentity elements in each factor. At every fixed density 1/3 < d < 1, their random free-product quotient, formed from independent uniform cyclically reduced syllable relators of exact length with replacement, has Kazhdan’s property (T) with probability tending to one through all integer lengths. This answers the property-(T) question in Problem 16 of the 2026 Oberwolfach report on median geometry. The proof completes model-specific steps of an earlier proposed syllable-block spectral program: a weighted endpoint calculation handles unequal alphabets, torsion, and all three length residues. It uses the established triangular-link criterion and matrix concentration. The conclusion includes fixed-radius word-ball alphabets and makes no uniform assertion for growing parameters.

The idea in plain English

Start with at least three groups, combine them in a free product, and impose many randomly chosen equations. The question is whether the resulting group usually has Kazhdan’s property (T), a strong form of rigidity: in unitary representations, almost-invariant vectors force a genuinely invariant vector.

For the precise sampling rule in this paper, the probability of property (T) approaches one whenever the fixed density is greater than one-third. Density controls the number of random equations as a power of the available relator count; it is not the fraction of all available equations that is sampled.

The result allows unequal finite syllable alphabets, torsion in the factors and every sufficiently large integer relator length. It answers the at-least-three-factor question in the 2026 Oberwolfach report. The new density range includes one-third < d ≤ one-half, since an earlier collapse theorem already covers d > one-half.

The general strategy was proposed earlier by Montee and Steenbock. This paper supplies the calculations and transfer argument needed for that strategy in the stated model, with explicit credit for earlier spectral criteria, block constructions and concentration inequalities.

The precise result

Fix n ≥ 3 nontrivial finitely generated groups Gᵢ and nonempty finite inverse-closed generating sets Bᵢ ⊂ Gᵢ ∖ {1}. For each ℓ ≥ 2, Sℓ consists of rooted syllable words of length exactly ℓ whose adjacent factors differ, including the last–first boundary. Words are not identified under rotation or inversion.

For a fixed density 1/3 < d < 1, sample ⌊|Sℓ|ᵈ⌋ independent uniform elements of Sℓ with replacement and quotient G₁ ∗ ⋯ ∗ Gₙ by their normal closure. Theorem 1 states that the probability of Kazhdan’s property (T) tends to one as ℓ tends to infinity through all integers.

The factors, alphabets and density remain fixed throughout the limit. Fixed-radius word-ball alphabets with positive integer radius are included, and factors may have torsion, be infinite or fail to be finitely presented.

The proof supplies a weighted endpoint kernel, deletion estimates for blocks equal to their inverses, spectra for all three length residues, concentration for independent whole-relator Laplacians, and a quotient onto a subgroup of index at most two.

Section 6 also treats the distinct sampling law of independent uniform linear reduced words, which omits the wrap-around condition, by retaining cyclic words at a slightly smaller density and then passing to a further quotient.

Why the result holds

A complete multipartite alphabet matrix counts legal syllable sequences. Its Perron eigenvector supplies the correct endpoint weights when the factors have different alphabet sizes. Blocks fixed by inversion are exponentially scarce and can be discarded.

Cut each relator into three almost-equal blocks. The mean link compresses exactly to a matrix indexed by block type and endpoints. Its limit has a spectral gap strictly greater than one-half for all three length residues.

Expected minimum degree grows exponentially when d > 1/3. Matrix concentration for the independent relator contributions and a separate degree bound transfer the mean spectral gap to the actual random link.

The classical triangular-link criterion gives property (T) to the block presentation. Carefully chosen common suffixes show that its image in the desired quotient has index at most two, so property (T) passes to the whole quotient.

Scope and limitations

The result requires at least three factors. It does not resolve the full two-factor version of Question 1.9 in Random Quotients of Free Products; for two factors the odd-length cyclic universe is empty.

There is no conclusion at d = 1/3, no finite-length guarantee, and no uniform estimate when the factors, alphabet sizes, ball radius or density vary with relator length. The one-third bound is sufficient, not claimed sharp.

The law is uniform sampling with replacement from rooted cyclically reduced syllable words of exact length. It must not be silently replaced by uniform conjugacy classes, ordinary free-group words, a different word metric, or sampling without replacement.

The theorem proves property (T), not infinitude or factor embeddings. Above one-half, known collapse already implies property (T); those facts and the models to which they apply require their own hypotheses.

The three-block spectral strategy, almost-equal blocks, common-suffix finite-index method, link criterion and matrix concentration have explicit predecessors. The contribution is their model-specific completion. No exhaustive novelty, historical firstness or universal Gromov-model threshold is claimed.

The finite verification routines test exact formulas and artifact integrity. They cannot certify the asymptotic theorem, property (T), external proof dependencies or priority.

Status and AI use

OpenAI tools were used extensively in solving the problem, developing and checking the proof, drafting, coding and adversarial verification. AI reviews and finite diagnostic checks do not constitute external human peer review or formal proof-assistant certification.

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Collaborate on this result

I welcome substantive collaboration on independent verification of the weighted link calculation, the torsion-compatible finite-index argument and the use of matrix concentration, as well as comparison with the random-group literature and preparation for journal review. Expertise in geometric group theory or spectral methods would be particularly useful.

Authorship and responsibilities would be discussed in relation to the substantive contributions made.

Cite this paper

Alec Kriebel. (2026). Property (T) above one-third density in random free-product quotients (Version 1.0.0) [Preprint]. Zenodo. https://doi.org/10.5281/zenodo.23390181

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Background and prior work

• Median Geometry and Applications, Oberwolfach Report 8/2026: Problem 16 on printed pp. 535–536 asks the property-(T) question for at least three factors. The paper also distinguishes the report’s linear-normal shorthand from the cyclic sampling law.
• Einstein, Krishna, Montee, Ng and Steenbock, Random Quotients of Free Products: Definition 1.1 fixes the word-ball syllable law. Theorem 1.2 gives collapse above one-half for at least three factors; Question 1.9 also includes two-factor parameters excluded here.
• Montee and Steenbock, Dynamical, geometric and analytic properties of random groups: The ESI Annual Report 2024, pp. 77–79, already proposes splitting relators into three syllable-respecting parts and leaves the spectral-gap and transfer questions open. The volume bears copyright 2025 and an August 2025 preface.
• Ashcroft, Property (T) in density-type models of random groups: Theorem B and Section 6 establish the earlier all-integer-length result using almost-equal blocks in the ordinary free-group model.
• Druţu and Mackay, Random groups, random graphs and eigenvalues of p-Laplacians: Lemma 12.2 supplies the common-suffix finite-index precedent; the present argument also avoids inverse-fixed blocks caused by torsion.
• Kotowski and Kotowski, Random groups and property (T): Żuk’s theorem revisited: Theorem 2.13 supplies the established link-gap criterion with multigraph multiplicities; Remarks 2.10–2.11 give the quotient and finite-index permanence properties.
• Tropp, User-friendly tail bounds for sums of random matrices: Corollary 5.2 and Remark 5.3 provide matrix Chernoff bounds. Independence is used between whole sampled relators, not between their three link edges.
• Oppenheim, Property (T) for random groups in the density model with d > 1/4: This separate result concerns the Gromov model along lengths divisible by four. It is context, not an input or a theorem for the present syllable law.