Injectivity of one-generator one-relator quotients of nonzero exponent sum over an arbitrary group (the strong form of the Kervaire conjecture), as posed in Freedman's 1978 remarks on first-degree equations in groups

In a 1978 paper, Michael Freedman developed a topological fixed-point framework for solving first-degree equations over groups: if a path-connected group of bounded isometries of a metric space contains a closed ball with the weak fixed point property, then every equation $z^{e_1}g_1\cdots z^{e_n}g_n=1$ of $z$-degree $\pm 1$ over it is solvable inside the group. The paper proves the criterion for connected Lie groups with bi-invariant metrics, exhibits an explicit unsolvable degree-$\pm 1$ equation in the connected solvable Lie group of $2\times 2$ real upper-triangular matrices with positive diagonal, constructs extensions realizing abelian isometric actions inside path-connected groups of bounded isometries, and derives a residual-finiteness obstruction for realizing arbitrary finitely generated groups in identity components of isometry groups of manifolds. The central issue the paper leaves open is algebraic: for an arbitrary group $G$ and any word $r\in G*\langle z\rangle$ whose exponent sum $\deg_z(r)$ is nonzero, must the natural map $G\to (G*\langle z\rangle)/\langle\!\langle r\rangle\!\rangle$ be injective? This is exactly the strong form of the Kervaire conjecture (also called the Kervaire–Laudenbach conjecture), a problem originating in Kervaire's 1965 classification of high-dimensional knot groups. It is known for residually finite, hyperlinear, locally indicable, and certain solvable coefficient groups (all nonzero degrees), for torsion-free groups in degree $\pm 1$, and for proper-power relators, but remains open in general as of 2026; a 2023–2024 claimed proof of the weak form by Kawauchi has not been accepted in the mainstream literature, which continues to treat the conjecture as open. The problem posed here is to prove or refute the injectivity conjecture in full generality.

Background

Equations over groups. Let $G$ be a group and let $z$ be a formal symbol. An equation over $G$ (with one unknown) is an expression $r(z)=1$ where $r$ is a word in $z$ and finitely many elements of $G$, i.e. an element of the free product $G*\langle z\rangle$, the group of reduced alternating words in elements of $G$ and powers of the infinite cyclic group $\langle z\rangle$. Solving the equation over $G$ means finding an element $z_0$ in some overgroup $K\supseteq G$ with $r(z_0)=1$ in $K$. Write $\langle\!\langle r\rangle\!\rangle$ for the normal closure of $r$ in $G*\langle z\rangle$ (the smallest normal subgroup containing $r$) and $\bar G=(G*\langle z\rangle)/\langle\!\langle r\rangle\!\rangle$. There is a standard dictionary: the equation $r(z)=1$ is solvable over $G$ if and only if the natural homomorphism $G\to\bar G$ is injective (a solution in an overgroup $K$ factors the quotient map through $K$, forcing $G\to\bar G$ to be injective; conversely $\bar G$ itself is such an overgroup when the map is injective). The exponent sum (or degree) $\deg_z(r)$ is the sum of the exponents of $z$ in $r$; when $\deg_z(r)\neq 0$ the equation is called nonsingular (or regular). The degree is the coarsest invariant: if $\deg_z(r)=0$ then $\bar G$ surjects onto $\mathbb{Z}$ (send $G$ to $0$ and $z$ to $1$), and if $|\deg_z(r)|=e\ge 2$ then $\bar G$ surjects onto $\mathbb{Z}/e\mathbb{Z}$, so $\bar G$ can be trivial only when $\deg_z(r)=\pm 1$.

The conjecture. In its weak form, the Kervaire conjecture asserts that for every nontrivial $G$ and every $r\in G*\langle z\rangle$ with $\deg_z(r)\neq 0$, the quotient $\bar G$ is nontrivial — adjoining one generator and one relation of nonzero degree cannot collapse a nontrivial group to the trivial group. In the language of the weight $w(H)$ of a group $H$ (the minimal number of elements needed to normally generate $H$), this says that $G*\mathbb{Z}$ has weight at least $2$ whenever $G\neq 1$. The problem under study is the strong form, also called the Kervaire–Laudenbach conjecture: for every group $G$ and every $r\in G*\langle z\rangle$ with $\deg_z(r)\neq 0$, the natural map $G\to\bar G$ is injective — equivalently, every nonsingular one-variable equation over an arbitrary group is solvable in some overgroup. The weak form concerns exactly the degree-$\pm 1$ nontriviality statement; the strong form genuinely concerns every nonzero degree, since no reduction of the degree is known for injectivity.

History. The question originates in Kervaire's 1965 work on the classification of high-dimensional knot groups, which hinges on which groups have weight one. Gerstenhaber and Rothaus (1962) proved, by degree theory of word maps on compact Lie groups (a finite group embeds in a unitary group $U(n)$, and a self-map of a compact manifold with nonzero degree is surjective), that nonsingular equations over finite — hence, by a residual-finiteness argument, over residually finite — coefficient groups are solvable. Freedman's 1978 paper recast solvability as a fixed-point problem and posed the injectivity conjecture explicitly, in both a finitely generated and an arbitrary-group form; a weaker statement contributed by Freedman is recorded on the 1970s Berkeley problem list compiled by Kirby (Problem 66); see Chen's 2025 paper.

Freedman's fixed-point framework (source paper). Call an isometry $g$ of a metric space $X$ bounded if $\sup_{x\in X} d_X(x,g(x))<\infty$. A path-connected group $G$ of bounded isometries carries the supremum metric $d_G(g,h)=\sup_x d_X(g(x),h(x))$; a closed ball $B_s$ has the weak fixed point property (wfpp) if every continuous self-map of $B_s$ that is homotopic within $B_s$ to a constant map has a fixed point. The paper's main theorem: if some ball $B_s\subset G$ has the wfpp, then every equation $r(z)=z^{e_1}g_1\cdots z^{e_n}g_n=1$ of $z$-degree $\pm 1$ over $G$ has a solution in $G$ itself, because $z\mapsto r(z)$ restricts to a continuous self-map of a suitable ball to which the wfpp applies. Left translations of a connected Lie group with a bi-invariant Riemannian metric are bounded isometries of displacement $d(e,g)$, so the criterion covers compact Lie groups (recovering the Gerstenhaber–Rothaus finite case) and, more generally, connected groups admitting bi-invariant metrics. The hypotheses cannot simply be dropped: in the connected solvable Lie group $B$ of real $2\times 2$ upper-triangular matrices with positive diagonal, the explicit equation $z^{-2}\begin{pmatrix}2&1\\ 0&1\end{pmatrix}z\begin{pmatrix}1&0\\ 0&2\end{pmatrix}=I$ of degree $-1$ has no solution — entry comparison forces $z$ to have both diagonal entries equal to $2$ and then yields $4x=4x+4$. The paper also proves an extension mechanism (every free abelian group of isometries of $X$ extends, via an iterated mapping-torus construction, to a path-connected group of bounded isometries of an enlarged metric space) and a manifold obstruction (by the Myers–Steenrod theorem the isometry group of a Riemannian manifold is a Lie group, so a finitely generated subgroup of its identity component has a residually finite adjoint image, by Malcev's theorem on finitely generated linear groups — blocking noncompact embedding theorems for non-residually-finite groups). Finally, the paper states, without proof, the injectivity conjectures in both forms — the problem under study — and derives a knot-theoretic consequence: if for a nontrivial knot $k$ the group $H_2(\pi_1(M_k))$ is nonzero, where $M_k$ is the 3-manifold of 0-framed surgery on $k$ (obtained by removing a tubular neighborhood of $k$ and regluing so that the meridian is identified with the longitude $\ell$; then $\pi_1(M_k)\cong \pi_1(S^3\setminus k)/\langle\!\langle\ell\rangle\!\rangle$), then one of three things happens: $M_k=S^1\times S^2$ (a failure of Property R — the statement that 0-surgery on a nontrivial knot never gives $S^1\times S^2$), $M_k=S^1\times S^2\#\Sigma^3$ with $\Sigma^3$ a homotopy 3-sphere counterexample to the Poincaré conjecture, or the injectivity conjecture fails for a finitely generated group (a quotient of the fundamental group of a homology 3-sphere).

Known cases and status. The conjecture (strong form) is known when: $G$ is residually finite (every nontrivial element survives in a finite quotient) — Gerstenhaber–Rothaus 1962; $G$ is locally indicable (every nontrivial finitely generated subgroup surjects onto $\mathbb{Z}$) — Howie 1981 and parallel early-1980s work; $G$ is hyperlinear or sofic (approximately embeddable in unitary groups with respect to the normalized Hilbert–Schmidt norm, or in finite symmetric groups with Hamming distance) — Pestov 2008, Klyachko–Thom 2017; $G$ is torsion-free and $\deg_z(r)=\pm 1$ — Klyachko 1993; $r$ is a proper power $w^m$, $m\ge 2$, $\deg_z(w)=1$, with $G$ arbitrary — Klyachko–Lurye 2012; for layered systems over torsion-free groups with exponent matrix invertible over $\mathbb{Z}$ — Forester–Rourke 2005; $G$ is solvable with a subnormal series with torsion-free abelian quotients — Klyachko–Mikheenko–Roman'kov 2024; and for subgroups of unit groups of Magnus power series rings — Ramirez-Côté–Wise 2025 (already a consequence of residual finiteness, but with a new fixed-point method). Counterexamples to the weak form are heavily constrained: the relator must have degree exactly $\pm 1$, and the coefficient group must be perfect (equal to its own commutator subgroup) with no proper normal subgroups of finite index (Freedman; Bogley–Edjvet–Williams), must have torsion, and can be chosen simple (Klyachko). No counterexample is known, and no proof is known for arbitrary groups; Chen's 2025 Trans. AMS paper states the conjectures "remain open in general" and that no significant breakthrough beyond Klyachko's theorem has been made in three decades.

Problem Statement

Prove or refute the following conjecture (the strong form of the Kervaire conjecture, also called the Kervaire–Laudenbach conjecture; it is the central open question posed in Freedman's 1978 paper, stated there both for finitely generated groups and for arbitrary groups).

Conjecture. For every group $G$ and every element $r$ of the free product $G*\langle z\rangle$ (where $\langle z\rangle$ is the infinite cyclic group generated by the formal symbol $z$), if the exponent sum $\deg_z(r)$ — the sum of the exponents of $z$ occurring in $r$ — is nonzero, then the natural homomorphism $G\longrightarrow (G*\langle z\rangle)/\langle\!\langle r\rangle\!\rangle,$ induced by the inclusion of the factor $G$ followed by the quotient by the normal closure of $r$, is injective.

Equivalently: every nonsingular (regular) one-variable equation over an arbitrary group — every equation $r(z)=1$ with coefficients in $G$ and $\deg_z(r)\neq 0$ — is solvable in some overgroup containing $G$.

Scope.

1. The finitely generated and the arbitrary-group forms of the conjecture are equivalent: if injectivity fails for some pair $(G,r)$, the killed element is a finite product of conjugates of $r^{\pm 1}$, and the subgroup of $G$ generated by that element, the coefficients of $r$, and the finitely many elements appearing in the conjugating words already exhibits the failure; so a proof (or refutation) for finitely generated groups is one for all groups.
2. The conjecture is not restricted to any proper class of coefficient groups (residually finite, hyperlinear or sofic, locally indicable, torsion-free, solvable, orderable, linear, etc.) or to any special shape of relator (proper powers, layered systems, invertible leading coefficients, etc.): it is asserted for every group $G$ and every $r\in G*\langle z\rangle$ with $\deg_z(r)\neq 0$.
3. Equivalently, the conjecture fails exactly when there exist a group $G$, a word $r\in G*\langle z\rangle$ with $\deg_z(r)\neq 0$, and an element $1\neq g\in G$ that lies in the normal closure of $r$ in $G*\langle z\rangle$.

Adjacent open questions from the source paper. Freedman's paper leaves two further unresolved issues that are separable from the conjecture above (neither is necessary nor sufficient for it): whether every finitely generated group acting by isometries on a metric space can be absorbed into the identity component of the group of bounded isometries of some enlarged metric space (the paper's proposed route to applying its fixed-point criterion, carried out for free abelian actions and obstructed for Riemannian manifolds); and the classification of connected non-compact Lie groups in which all first-degree equations are solvable. The paper's knot-theoretic consequence is a one-way bridge rather than an additional deliverable: given Gabai's Property R theorem and the proved Poincaré conjecture, a nontrivial knot whose 0-surgery group has nonzero second homology would refute the conjecture, and the conjecture implies this homology group always vanishes.

References

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