Finite groups in the property-T collapse window

Definitions and mathematical statement

Let $W_n$ be the reduced words of length $n$ over $a,a^{-1},b,b^{-1}$; $|W_n|=4\cdot3^{n-1}$. For a density $d$, the source samples $3^{nd}$ independent uniform relators from $W_n$; an integer rounding convention is needed. Given the resulting sequence $R_1,\ldots,R_m$, define the quotient chain \[ G_k=\langle a,b\mid R_1,\ldots,R_k\rangle\qquad(0\le k\le m). \] On a realization containing a trivial quotient, put \[ \tau_0=\min\{k:G_k=1\},\qquad \tau_\infty=\max\{k<\tau_0:G_k\text{ is infinite}\}. \] The count requested is \[ N_n=\left|\{[G_k]:\tau_\infty<k<\tau_0,\ 1<|G_k|<\infty\}\right|, \] where brackets mean abstract group isomorphism class. Repeated unchanged quotients are counted once, not once per added relator.

The question asks for a useful upper bound on this finite-window count and conjectures an almost-sure bound uniform in $n$. On a specified joint probability space, one possible interpretation is \[ \Pr\!\left(\sup_nN_n<\infty\right)=1. \] A deterministic uniform constant, a random almost-sure bound, and tightness/high-probability bounds are distinct; the source does not choose between them or specify the coupling across $n$.

There is an essential parity qualification. If $n$ is even, sending both $a,b$ to $1\in{\mathbb{Z}}/2{\mathbb{Z}}$ kills every length-$n$ relator, so every $G_k$ still surjects onto ${\mathbb{Z}}/2{\mathbb{Z}}$. Such a chain cannot collapse to the trivial group. Also a fixed finite sample of relators need not contain a collapse at its chosen density. Restricting to odd lengths, changing the terminal group, conditioning on collapse, or extending an infinite relator stream are different repairs; none is silently adopted. Until these choices are supplied, $N_n$ is only conditionally defined by the displayed stopping indices.

The supplied summary reports no verified finite-window census bound and distinguishes related random-group asymptotic results from this sequential count. The title's property-T motivation is retained without adding a property-T hypothesis or a proof of the conjecture.

Short English statement

Bound the number of distinct finite nontrivial quotients in the sequential random-relator collapse window, ideally almost surely uniformly in word length. Even-length parity, the sample horizon, and the meaning of the joint almost-sure bound need clarification.

Sources

• Local UnsolvedMath registry, permanent ID $10000002$, exact source code catalogue entry, "Finite groups in the property-T collapse window." Proposer Itai Benjamini, 2007.
• Benjamini, A 1'st order phase transition via property T, May 2007, Question 1, one-page note; https://www.wisdom.weizmann.ac.il/~itai/hplast.pdf. Recorded access 2026-07-29; manual transcription checked against archived text.
• Recorded preservation copy: https://arquivo.pt/wayback/20201231041601id_/https://www.wisdom.weizmann.ac.il/~itai/hplast.pdf.
• Supplied OPEN-TRIAGE summary mentions Ollivier and Kotowski--Kotowski as related density-model literature, but provides no exact theorem or primary citation resolving the finite-window count.