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Math-2191 · mathematics · Algebra

Sofic Groups

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Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
Is every group sofic? Here soficity means that every finite subset admits arbitrarily accurate approximations by permutations of a finite set: multiplication is respected on all but a prescribed small proportion of points, and distinct group elements are represented by permutations differing on nearly all points.

Formulation references: [S1], [S2].

Short English statement

Is every group sofic? Here soficity means that every finite subset admits arbitrarily accurate approximations by permutations of a finite set: multiplication is respected on all but a prescribed small proportion of points, and distinct group elements are represented by permutations differing on nearly all points.

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 2

Math-2191-A2 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2191

The main finite-field one-sided-inverse pair produces a nonsurjunctive group. Soficity entails surjunctivity and direct finiteness for the relevant finite-field algebra, so this same finitely generated group is not sofic. For the integral matrix of Theorem 1.1, the Fuglede--Kadison determinant belongs to $(0,1)$. Sofic groups satisfy the determinant conjecture for every integral group-ring matrix, so the example’s group cannot be sofic. Within these assumptions, this is a claimed negative answer to the stated question.

The Sofic Groups Conjecture

OpenAI's claim

The linked manuscripts present the following claims by OpenAI: A finite characteristic-two field and a finitely presented group are claimed to produce a scalar one-sided inverse pair; the resulting shift is injective without being onto and the group is nonsofic; For a particular odd prime $p$, the authors claim a one-sided inverse defect over a field with $p^4$ elements, together with a nonsurjective injective shift; An integral group-ring matrix is claimed to be rationally invertible while its Fuglede--Kadison determinant lies strictly between $0$ and $1$; Another construction claims a torsion-free, finitely presented $G$ for which $\mathbb F_2[G]$ contains $a,b$ satisfying $ab=1$ and $ba\ne1$. Together they give a claimed negative answer in the matching scope.

Mathematical statement of the claimed result

Result 1.
Take a finite field $K$ with $\operatorname{char}K=2$. The constructed group $G$ is finitely presented and has an element of odd prime order. Two finite group-ring sums meet \[ a_{\mathrm{out}},b_{\mathrm{out}}\in K[G],\qquad a_{\mathrm{out}}b_{\mathrm{out}}=1,\qquad b_{\mathrm{out}}a_{\mathrm{out}}\ne1. \] Its specification uses terminating finite-set and finite-field computations. Unlike the torsion-free companion, this particular group contains torsion.
Result 2.
Define the integers \[ m=\binom{1200}{600},\qquad p=\min\{h\in\mathbb Z:h>1,\ h\mid(m!)^2+1\}. \] The claimed example has $p$ an odd prime, $|K|=p^4$, and a specified finitely generated group $G$ containing torsion. Its finite sums $a,b\in K[G]$ satisfy \[ ab=1\quad\hbox{and}\quad ba\ne1. \] This is an example in a specified odd characteristic, rather than a theorem for each odd prime.
Result 3.
For some finitely generated discrete group $G_D$, integer $n\ge1$, and matrix \[ A\in M_n(\mathbb Z[G_D]),\qquad A\text{ invertible over }\mathbb Q[G_D], \] the operator induced by the left regular action $T_A$ on $\ell^2(G_D)^n$ has Fuglede--Kadison determinant \[ 0<\det_{\mathcal N(G_D)}(T_A)<1. \] Here $\mathcal N(G_D)$ is the group von Neumann algebra and the logarithmic spectral integral defining the determinant is finite and negative. The characteristic-two direct-finiteness result quoted in the proof is an input, not an additional determinant hypothesis. The convention is $(L_g\xi)(h)=\xi(g^{-1}h)$, and the matrix trace $\tau_n$ is unnormalized. For the normalized spectral law $\mu_A$ of $T_A^*T_A$, \[ \log\det_{\mathcal N(G_D)}(T_A)=\tfrac12\tau_n(\log(T_A^*T_A))=\tfrac n2\int\log t\,d\mu_A(t). \]
Result 4.
The construction supplies a torsion-free, finitely presented group $G$, with a finite two-dimensional $K(G,1)$, and elements of its characteristic-two group algebra satisfying \[ a,b,c\in\mathbb F_2[G],\qquad ab=1,\quad ac=0,\quad c\ne0,\quad ba\ne1. \] In particular, $a\ne0$ and $e=ba$ obeys $e^2=e$, $e\notin\{0,1\}$. Thus this same example supplies failures of direct finiteness, absence of zero divisors, and triviality of scalar idempotents. These assertions concern $\mathbb F_2[G]$; the separate integral-unit assertion is not a consequence.

Outline of the argument

Incidence data and polynomial pairings first give a defective inverse pair of matrices. A geometric group-embedding construction preserves the defect, and character packing realizes the matrix algebra inside a larger group algebra; dual convolution gives the cellular-automaton consequence. The construction uses translation representations of finite groups: a high-degree subspace is killed on one side and a weighted pairing supplies the other annihilation. Finite group identifications and a quadratic field tower package these local relations into the scalar inverse pair; the claim concerns the specified prime rather than all odd characteristics.

A characteristic-two inverse defect is lifted using finite-coordinate projections in a wreath-product setting. These projections detect a selected coefficient, and a tracial logarithmic derivative applied to block shears calculates a determinant below the conjectured integral lower bound. An unequal-side parity pattern supplies the one-sided inverse relation in labeled graphs. Weighted counts handle repeated edge traversals, and a separator reduces arbitrarily large planar arrangements to bounded path collections; cone attachment and root separation then preserve the inverse defect while proving torsion-freeness.

Formal verification

For family 197, the documented Lean entries include the scalar inverse defects in the specified characteristics and an operator-invertible determinant counterexample. The characteristic-two search recipe is included, but the nonsofic consequence, the extra spectral-integral conclusion, and the later torsion-free construction are outside the selected statements. The listed Comparator files are GroupRingDeterminant.lean, KaplanskyDirectFiniteness.lean, KaplanskyFinitelyPresented.lean, OddKaplansky.lean. Named declarations include OAI.GroupRingDeterminant.main, OAI.KaplanskyCounterexample.finitelyPresented_counterexample, OAI.OddKaplansky.main_theorem. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

Sources

Go to contribution →

Math-2191-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2191

The main finite-field one-sided-inverse pair produces a nonsurjunctive group. Soficity entails surjunctivity and direct finiteness for the relevant finite-field algebra, so this same finitely generated group is not sofic. For the integral matrix of Theorem 1.1, the Fuglede--Kadison determinant belongs to $(0,1)$. Sofic groups satisfy the determinant conjecture for every integral group-ring matrix, so the example’s group cannot be sofic. Within these assumptions, this is a claimed negative answer to the stated question.

Sofic Groups

OpenAI's claim

The linked manuscripts present the following claims by OpenAI: A finite characteristic-two field and a finitely presented group are claimed to produce a scalar one-sided inverse pair; the resulting shift is injective without being onto and the group is nonsofic; For a particular odd prime $p$, the authors claim a one-sided inverse defect over a field with $p^4$ elements, together with a nonsurjective injective shift; An integral group-ring matrix is claimed to be rationally invertible while its Fuglede--Kadison determinant lies strictly between $0$ and $1$; Another construction claims a torsion-free, finitely presented $G$ for which $\mathbb F_2[G]$ contains $a,b$ satisfying $ab=1$ and $ba\ne1$. Together they give a claimed negative answer in the matching scope.

Mathematical statement of the claimed result

Result 1.
Take a finite field $K$ with $\operatorname{char}K=2$. The constructed group $G$ is finitely presented and has an element of odd prime order. Two finite group-ring sums meet \[ a_{\mathrm{out}},b_{\mathrm{out}}\in K[G],\qquad a_{\mathrm{out}}b_{\mathrm{out}}=1,\qquad b_{\mathrm{out}}a_{\mathrm{out}}\ne1. \] Its specification uses terminating finite-set and finite-field computations. Unlike the torsion-free companion, this particular group contains torsion.
Result 2.
Define the integers \[ m=\binom{1200}{600},\qquad p=\min\{h\in\mathbb Z:h>1,\ h\mid(m!)^2+1\}. \] The claimed example has $p$ an odd prime, $|K|=p^4$, and a specified finitely generated group $G$ containing torsion. Its finite sums $a,b\in K[G]$ satisfy \[ ab=1\quad\hbox{and}\quad ba\ne1. \] This is an example in a specified odd characteristic, rather than a theorem for each odd prime.
Result 3.
For some finitely generated discrete group $G_D$, integer $n\ge1$, and matrix \[ A\in M_n(\mathbb Z[G_D]),\qquad A\text{ invertible over }\mathbb Q[G_D], \] the operator induced by the left regular action $T_A$ on $\ell^2(G_D)^n$ has Fuglede--Kadison determinant \[ 0<\det_{\mathcal N(G_D)}(T_A)<1. \] Here $\mathcal N(G_D)$ is the group von Neumann algebra and the logarithmic spectral integral defining the determinant is finite and negative. The characteristic-two direct-finiteness result quoted in the proof is an input, not an additional determinant hypothesis. The convention is $(L_g\xi)(h)=\xi(g^{-1}h)$, and the matrix trace $\tau_n$ is unnormalized. For the normalized spectral law $\mu_A$ of $T_A^*T_A$, \[ \log\det_{\mathcal N(G_D)}(T_A)=\tfrac12\tau_n(\log(T_A^*T_A))=\tfrac n2\int\log t\,d\mu_A(t). \]
Result 4.
The construction supplies a torsion-free, finitely presented group $G$, with a finite two-dimensional $K(G,1)$, and elements of its characteristic-two group algebra satisfying \[ a,b,c\in\mathbb F_2[G],\qquad ab=1,\quad ac=0,\quad c\ne0,\quad ba\ne1. \] In particular, $a\ne0$ and $e=ba$ obeys $e^2=e$, $e\notin\{0,1\}$. Thus this same example supplies failures of direct finiteness, absence of zero divisors, and triviality of scalar idempotents. These assertions concern $\mathbb F_2[G]$; the separate integral-unit assertion is not a consequence.

Outline of the argument

Incidence data and polynomial pairings first give a defective inverse pair of matrices. A geometric group-embedding construction preserves the defect, and character packing realizes the matrix algebra inside a larger group algebra; dual convolution gives the cellular-automaton consequence. The construction uses translation representations of finite groups: a high-degree subspace is killed on one side and a weighted pairing supplies the other annihilation. Finite group identifications and a quadratic field tower package these local relations into the scalar inverse pair; the claim concerns the specified prime rather than all odd characteristics.

A characteristic-two inverse defect is lifted using finite-coordinate projections in a wreath-product setting. These projections detect a selected coefficient, and a tracial logarithmic derivative applied to block shears calculates a determinant below the conjectured integral lower bound. An unequal-side parity pattern supplies the one-sided inverse relation in labeled graphs. Weighted counts handle repeated edge traversals, and a separator reduces arbitrarily large planar arrangements to bounded path collections; cone attachment and root separation then preserve the inverse defect while proving torsion-freeness.

Formal verification

For family 197, the documented Lean entries include the scalar inverse defects in the specified characteristics and an operator-invertible determinant counterexample. The characteristic-two search recipe is included, but the nonsofic consequence, the extra spectral-integral conclusion, and the later torsion-free construction are outside the selected statements. The listed Comparator files are GroupRingDeterminant.lean, KaplanskyDirectFiniteness.lean, KaplanskyFinitelyPresented.lean, OddKaplansky.lean. Named declarations include OAI.GroupRingDeterminant.main, OAI.KaplanskyCounterexample.finitelyPresented_counterexample, OAI.OddKaplansky.main_theorem. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

Sources

Go to contribution →