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Math-2405 · mathematics · Group theory

Solvability of Group Equations and Surjectivity of Word Maps

Status Partially solved · unsolvedmath.comLLM status Open unverified

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.
Can every nonsingular equation $w\in\Gamma*F_n$ with coefficients in a group $\Gamma$ be solved over $\Gamma$? Also, for every nontrivial word $w\in F_2$, does there exist $n_0$ such that the word map $w:\operatorname{PU}(n)\times\operatorname{PU}(n)\to\operatorname{PU}(n)$ is surjective for every $n\geq n_0$?

Formulation references: [S1], [S2].

Short English statement

Can every nonsingular equation $w\in\Gamma*F_n$ with coefficients in a group $\Gamma$ be solved over $\Gamma$? Also, for every nontrivial word $w\in F_2$, does there exist $n_0$ such that the word map $w:\operatorname{PU}(n)\times\operatorname{PU}(n)\to\operatorname{PU}(n)$ is surjective for every $n\geq n_0$?

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2405-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2405

The October theorem, numbered Theorem 1, solves nonsingular systems in coefficient-preserving overgroups. Its companion handles the stated one-equation subclass. Neither assertion supplies solutions inside the original compact group or surjectivity of the independent projective-unitary word maps, so that separate bundled clause remains unanswered.

Solvability of Group Equations and Surjectivity of Word Maps

OpenAI's claim

OpenAI presents a mathematical result relevant to this question in the selected manuscript. Any finite equation system over a group with full-rank exponent-sum matrix is claimed simultaneously solvable after embedding into a larger group.

Mathematical statement of the claimed result

Result 1.
Let $G$ be any group, $1\le m\le n$, and $w_i\in G*F_n$. For the exponent-sum matrix $E=(e_{ij})$, where $e_{ij}$ counts the total exponent of the $j$-th free generator in $w_i$, assume $\operatorname{rank}_{\mathbb Q}E=m$. The source claims injectivity of \[ G\longrightarrow (G*F_n)/\langle\!\langle w_1,\ldots,w_m\rangle\!\rangle. \] Equivalently, all equations $w_i=1$ can be solved simultaneously in an overgroup without identifying distinct coefficient elements of $G$.

Outline of the argument

Simultaneous unitary fixed-space constructions are measured by a traced spectral phase. Subadditivity and weighted-cycle calculations rule out the planar coefficient obstruction associated with a nonsingular system, allowing the coefficient group to embed in the solution overgroup.

Formal verification

For family 256, the selected statement is coefficient injectivity for a single relator of exponent sum $1$ or $-1$ in $A*\mathbb Z$. It underlies the Kervaire consequence but does not include the newer general nonsingular-system theorem. No Comparator is assigned to the selected manuscript here; the family page selects other papers or supporting statements. 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

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