Hyperlinearity of Every Group

Hyperlinearity of Every Group

Definitions and mathematical statement

Let $U(d)=\{U\in M_d({\mathbb{C}}):U^*U=I\}$ and $\|A\|_{2,d}=(d^{-1}\operatorname{Tr}(A^*A))^{1/2}$. A group $G$ is hyperlinear in the selected convention if, for every finite $F\subseteq G$ and $\varepsilon>0$, there is a map $\phi:F\to U(d)$ for some $d$ such that \[ \|\phi(g)\phi(h)-\phi(gh)\|_{2,d}<\varepsilon, \qquad \|\phi(g)-I\|_{2,d}>\sqrt2-\varepsilon\quad(g\ne e). \] The first condition applies only when $g,h,gh\in F$, and the second only to elements of $F$. Dimension may depend on $F$ and $\varepsilon$; exact homomorphisms are not required. Determine whether every group admits these finite approximate matrix models.

Formulation and concept references: [S1].

Short English statement

Can every finite portion of every group be modeled by unitary matrices that nearly preserve multiplication while keeping nonidentity elements detectably away from the identity?

Research attempt

Current frontier.

Dogon's Theorems 1.3 and 1.6 give non-hyperlinearity of suitable nonsplit central extensions under a flexible Hilbert--Schmidt stability, or weaker ucp-stability, hypothesis. Property (T) alone is not that hypothesis [S1]. A manuscript by Jihao Liu, in its consulted 21 September 2026 version, claims an explicit non-hyperlinear wreath product, using internality and normalization of commutants in matrix ultraproducts [E1]. Editorial status note: the original catalogue text is retained. That manuscript is a current proof claim, not an independently verified theorem in this notebook. The calculations below check a specific obstruction and its compatibility with the printed approximation convention; they do not audit its full analytic argument.

Attack route.

Three routes have rather different missing inputs. The central-extension route needs a stability theorem strong enough to invoke Dogon. A route through nonembeddable operator algebras needs the obstruction to concern a group's canonical trace, rather than just some trace on an algebra. The route pursued here instead tries to force two distinct lamps in a permutation wreath product to have the same image in every matrix ultraproduct. Its precise missing input is normalization of a subgroup commutant. This gives a useful finite-error question, even without assuming that the recent claim has been verified.

Attempt: an explicit group and a nontrivial word.

Use the construction in [E1], Theorem 1.1: \[ R_+={\mathbb{F}}_2[x_1,x_2,x_3],\qquad R={\mathbb{F}}_2[x_1^{\pm1},x_2^{\pm1},x_3^{\pm1}],\qquad \Gamma=\operatorname{EL}_3(R_+), \] \[ G=\operatorname{EL}_3(R)\rtimes_\alpha\operatorname{SL}_3({\mathbb{Z}}),\qquad W=\left(\bigoplus_{G/\Gamma}C_2\right)\rtimes G. \] Here $\operatorname{EL}_3(B)$ is generated by $I_3+bE_{ij}$, $i\ne j$, and $\alpha_A(x^v)=x^{Av}$ for column vectors $v\in{\mathbb{Z}}^3$. Write $a_q$ for the nonidentity lamp at the left coset $q$. Set \[ t=(I_3,I_3+E_{12}),\qquad \gamma=(I_3+x_2E_{12},I_3),\qquad x=a_{t\Gamma},\quad y=\gamma x\gamma^{-1},\quad b=yx. \] The following verifications are supplied here rather than inferred from the paper's claimed conclusion. The inverse monomial shear sends $x_2$ to $x_1^{-1}x_2$, so \[ t^{-1}\gamma t=(I_3+x_1^{-1}x_2E_{12},I_3)\notin\Gamma. \] Every matrix in $\operatorname{EL}_3(R_+)$ has polynomial entries, which justifies the last exclusion. Thus $\gamma t\Gamma\ne t\Gamma$. The lamps at these two cosets are independent involutions, so $x^2=y^2=e$ and $b\ne e$. Notice also that $t\Gamma t^{-1}\subseteq\Gamma$, since the forward shear sends polynomial monomials to polynomial monomials.

Let $M$ be a tracial matrix ultraproduct and $\rho:W\to U(M)$ a homomorphism. Put $D=\rho(\Gamma)'\cap M$. We isolate the following input, without claiming to establish it: \begin{equation}\label{wsref-0-65-71-31-35-34-2d-6e-6f-72-6d-61-6c-69-7a-65} \rho(t)D\rho(t)^*=D \quad\text{for every such }M\text{ and }\rho. \end{equation} Conditional lamp lemma. Equation (eq154-normalize) implies $\rho(b)=I$ for every $\rho$.

Proof. The lamp $a_\Gamma$ commutes with $\Gamma$, since this subgroup fixes its coset. Hence $\rho(a_\Gamma)\in D$. Normalization gives $X:=\rho(x)=\rho(t)\rho(a_\Gamma)\rho(t)^*\in D$. In particular $X$ commutes with $\rho(\gamma)$. Consequently $\rho(y)=X$, and $\rho(b)=X^2=I$. This uses an actual equality of commutants, not merely a one-sided inclusion. $\square$

The recent manuscript asserts centralizer results intended to supply this normalization in its wreath-product application; see its Theorems 6.7, 7.4, and 8.3 [E1]. We keep (eq154-normalize) as an unproved input in the present argument.

Attempt: a dimension-free finite-error obstruction.

The lamp argument admits an explicit estimate that does not require ultraproduct terminology. Let $F$ contain \[ e,\ \gamma,\ \gamma^{-1},\ x,\ \gamma x,\ y,\ b, \] and suppose $\phi:F\to U(d)$ satisfies the printed multiplicative inequalities with tolerance $\varepsilon$. Put $H=\phi(\gamma)$ and $X=\phi(x)$, and use normalized Hilbert--Schmidt norms throughout. All the products needed below have their factors and product in $F$. From $e^2=e$ and $\gamma\gamma^{-1}=e$ we obtain \[ \|\phi(e)-I\|_2<\varepsilon,\qquad \|\phi(\gamma^{-1})-H^*\|_2<2\varepsilon. \] Using $(\gamma x)\gamma^{-1}=y$, followed by $x^2=e$ and $yx=b$, gives \[ \|\phi(y)-HXH^*\|_2<4\varepsilon,\qquad \|X^2-I\|_2<2\varepsilon, \] and therefore \begin{equation}\label{wsref-0-65-71-31-35-34-2d-65-72-72-6f-72} \|\phi(b)-I\|_2 <7\varepsilon+\|HX-XH\|_2 \le 7\varepsilon+2\operatorname{dist}_2(X,\{H\}'). \end{equation} For the last inequality, subtract any matrix commuting with $H$ and use unitary invariance of the norm. No positivity or unitarity is required of that comparison matrix. The separation requirement for $b\ne e$ now forces \begin{equation}\label{wsref-0-65-71-31-35-34-2d-64-69-73-74-61-6e-63-65} \operatorname{dist}_2(X,\{H\}')> \frac{\sqrt2-8\varepsilon}{2}. \end{equation} In particular, any proposed uniform repair argument must overcome an order-one distance, not just improve a word-error constant. Equations (eq154-error)--(eq154-distance) are unconditional finite-matrix lemmas proved here; they do not assert that the distance is small.

Attempt: the repair estimate that really is available.

Suppose $\pi:\Lambda\to U(d)$ is an exact representation and a finite set $S\subset\Lambda$ has an adjoint spectral gap $\kappa>0$, in the following precise sense: \[ \sum_{s\in S}\|\pi(s)V\pi(s)^*-V\|_2^2 \ge\kappa^2\|V\|_2^2 \quad\text{whenever }V\perp\pi(\Lambda)'. \] Let $E$ be orthogonal projection onto the commutant in the Hilbert space $M_d({\mathbb{C}})$ with normalized trace inner product. Applying the gap to $A-E(A)$ proves \begin{equation}\label{wsref-0-65-71-31-35-34-2d-67-61-70} \|A-E(A)\|_2\le\kappa^{-1} \left(\sum_{s\in S}\|[A,\pi(s)]\|_2^2\right)^{1/2}. \end{equation} Indeed, $E(A)$ is fixed by conjugation and the orthogonal complement is invariant. For unitary $A$, this projection is also the trace-preserving conditional expectation onto the finite-dimensional commutant; hence $Y=E(A)$ is a contraction. Extend the polar partial isometry of $Y$ to a unitary $U$ in each matrix block of the commutant. Such an extension exists because initial and final ranks agree in every square block. Then \[ \|Y-U\|_2^2=\operatorname{tr}_d(1-|Y|)^2 \le1-\|Y\|_2^2=\|A-Y\|_2^2, \] so $\|A-U\|_2\le2\|A-E(A)\|_2$. This proves a concrete commuting-unitary repair theorem under the stated coordinate-level gap hypothesis.

It does not prove (eq154-normalize). Lifts of an ultraproduct representation need only be approximate representations in each coordinate. The spectral-gap inequality above has not been proved for those lifts, and replacing them by exact representations would import a stability premise of the kind isolated in [S1].

Stress test.

First, almost commuting need not mean close to the exact coordinate commutant. Take \[ U_n=\begin{pmatrix}1&0\\0&e^{i/n}\end{pmatrix},\qquad V=\begin{pmatrix}0&1\\1&0\end{pmatrix}. \] Then $\|[U_n,V]\|_{2,2}=|1-e^{i/n}|\to0$, whereas the commutant of $U_n$ is diagonal and $\operatorname{dist}_{2,2}(V,\{U_n\}')=1$. The limiting unitary is $I$, whose commutant is much larger. This is not a counterexample to a property-(T) theorem; it disproves an unjustified passage from limiting commutation to exact coordinate commutants.

Second, a trace-preserving one-sided inclusion need not be equality in an infinite finite von Neumann algebra. For example, inside the bilateral infinite tensor product $\overline\bigotimes_{k\in{\mathbb{Z}}}M_2$, the shift sends $D_+=\overline\bigotimes_{k\ge0}M_2$ properly into itself. In the crossed product by the shift this map is implemented by a unitary and preserves the trace. Thus a finite-dimensional dimension-counting argument cannot simply be transferred to the ultraproduct. This example is not an asserted model of the specified property-(T) construction.

Finally, we check the exact convention in the catalogue. Assume that the printed finite approximations exist for the countable group $W$. Choose increasing finite sets exhausting $W$, and approximations with $\varepsilon_n\to0$. Extend each map arbitrarily by $I$ outside its domain. The classes of these bounded unitary sequences in any nonprincipal tracial matrix ultraproduct give an exact homomorphism $\rho$. The elementary estimate on $\phi(e)$ above gives $\rho(e)=I$; no additional normalization was assumed. Moreover \[ \|\rho(b)-I\|_2=\lim_{n\to\omega}\|\phi_n(b)-I\|_2\ge\sqrt2. \] This contradicts the conditional lamp lemma. Thus (eq154-normalize), if justified, really would disprove the exact statement here, not merely a stronger pairwise-separation variant.

Outcome.

Outcome: Conditional reduction.

The supplied argument proves nontriviality of the candidate word, the finite-error obstruction (eq154-error), and the implication from commutant normalization to failure of the catalogue's approximations. It also proves a repair lemma for exact representations and exhibits why its naive use on approximate representations fails. These deductions are not a new construction of the underlying group, and no priority is claimed for the conditional lamp mechanism.

The unresolved step in this notebook is the uniform normalization input (eq154-normalize); the recent manuscript claims to provide it, but its full proof has not been independently verified here. Establishing that input would produce a negative answer and would also rule out soficity of this group, since permutation-matrix approximations are a special source of hyperlinear approximations [S1], Introduction. The weaker finite-matrix estimates alone do not supply a counterexample.

Sources

• S1 A. Dogon, Flexible Hilbert-Schmidt Stability versus Hyperlinearity for Property (T) Groups, arXiv:2211.10492v3 (25 August 2023). https://arxiv.org/html/2211.10492v3. Catalogue formulation source.
• E1 Jihao Liu, Nonhyperlinear groups exist, author-hosted manuscript, first posted 20 September 2026; consulted PDF dated 21 September 2026, Theorems 1.1--1.3, 6.7, 7.4, and 8.3. https://jihaoliu.org/ai-results/nonhyperlinear-groups-exist-2026-09-20.pdf. Consulted 27 September 2026. A proof claim; not treated as independently verified. The explicit group is attributed to this source.