The Kervaire theorem for groups

1 Introduction

Adding one generator and one relation to a nontrivial group should leave a nontrivial group. This is the Kervaire conjecture. We prove it by establishing coefficient injectivity in the case where the new generator has exponent sum \pm1.

Let A be a group, let \langle t\rangle\cong\Z, and set H=A*\langle t\rangle. Write p:H\to\Z for the homomorphism that kills A and sends t to 1. A word w\in H is unimodular if p(w)=\pm1. Denote its normal closure in H by \langle\!\langle w\rangle\!\rangle_H.

The equation w(t)=1 is solvable over A if there are an embedding A\to B and an element b\in B such that evaluating t at b gives w(b)=1. This is equivalent to injectivity of the coefficient map A\to H/\langle\!\langle w\rangle\!\rangle_H: every solution factors through this quotient, while an injective coefficient map lets the quotient itself supply a solution.

No restriction on torsion, cardinality, or presentation of A is imposed. The nontriviality assertion for all exponent sums follows at once.

1.1 Context and methods

Kervaire’s characterization of high-dimensional knot groups includes normal generation by a single element [8, Theorem 1.1]. Chen [2, preprint Section 1] traces the group-theoretic Kervaire problem to this setting. Its stronger Kervaire–Laudenbach form asks for coefficient injectivity whenever p(w)\ne0; such one-variable equations are called nonsingular. Theorem 1.1 proves its unimodular case, while Corollary 1.2 resolves the full nontriviality conjecture. At the level of assertions for all groups, this nontriviality statement is equivalent to unimodular coefficient injectivity: Klyachko records the standard reduction through an embedding into a simple group [11, preprint Section 3, Proposition 2]. We prove injectivity directly.

Gerstenhaber and Rothaus proved solvability of nonsingular systems of equations over finite groups in finite overgroups [3, Theorem 2]. In particular their theorem gives the required injectivity for finite coefficient groups, and hence for residually finite groups by separating each nonidentity coefficient in a finite quotient. Pestov records the resulting ultraproduct extension: nonsingular equations over hyperlinear groups have solutions in hyperlinear overgroups [15, preprint Corollary 10.4]. Hyperlinear groups admit embeddings into metric ultraproducts of unitary groups with normalized Hilbert–Schmidt metrics. Klyachko proved unimodular injectivity for torsion-free groups [10, Theorem 4, exponent-one case]. Chen gave a new proof through estimates for the complexity of surfaces [2, preprint Theorem 6.9]. For arbitrary coefficient groups, Klyachko and Lurye proved injectivity after imposing w^m=1 when w is unimodular and m\geq2 [12, preprint Section 1, Theorem; Section 5]; this does not impose w=1.

Kawauchi has published a proposed resolution of the universal nontriviality assertion through knot exteriors, relative collapses, and ribbon sphere-link groups [7, Theorem 1 and Corollary 2]. By the equivalence just recalled, its endpoint includes unimodular injectivity. The argument below takes a different route through spectral phase and a finite-dimensional fixed-space theorem; neither the collapse construction nor the ribbon-asphericity assertions enter its proof. Recent work of Klyachko, Mikheenko, and Olshanskii also distinguishes solvability inside a coefficient group from solvability in an overgroup [13, Sections 1 and 3]. Our conclusion concerns the latter and does not assert that a solution already lies in A.

Unitary groups and degree theory already play a central role in the Gerstenhaber–Rothaus method [3, Theorem 1]; see also [9, Section 1.1]. Corner-labelled diagrams and Euler-characteristic estimates likewise have a substantial history in equations over groups [12, preprint Sections 3–4] and [2, preprint Sections 2–4]. Here these methods meet through a trace comparison for permutations weighted by the left regular representation of A. The regular representation is available for every group and supplies a faithful finite trace without an approximation hypothesis on A. For the spectral ingredient, Thompson’s matrix exponential formula and its extensions provide a related precedent [1, Section 2 and Theorem 4.4]; Section 2 distinguishes those formulas from the unrestricted phase inequality proved here.

Two ingredients are useful independently of the diagram argument. The first is subadditivity of the trace of the fractional spectral phase of a unitary. The second is a fixed-space lemma: given k\geq1, signs s_1,\ldots,s_n\in\{1,-1\} with \sum_i s_i=\pm1, and P\in\U(nk), some X\in\U(k) makes \diag(X^{s_1},\ldots,X^{s_n})P have at least k fixed directions. We prove the latter by a degree calculation on a compact manifold that records a unitary together with a fixed k-plane. Their combination gives the planar obstruction below.

1.2 Proof overview

Suppose a coefficient g\in A becomes trivial after imposing w=1. A finite expression for g as a product of conjugates of w and w^{-1} can be drawn on a punctured disk. Pairing the occurrences of t produces disjoint bands between word disks. The complementary boundary curves read words in A.

The central assertion is Theorem 4.1: in a connected planar surface made from these disks and bands, one boundary word cannot be the only nontrivial one. To prove it, we encode travel along the boundaries by a weighted permutation. We measure that operator by the trace of its spectral phase. Mixing positive and negative word disks with a suitable unitary gives an upper bound. A second calculation, one boundary circle at a time, approaches that bound plus the nonnegative phase of the exceptional word. The Euler characteristic formula identifies the common term in these two calculations. The upper bound forces the extra term to vanish, so this word is trivial.

Section 2 proves the phase inequality, and Section 3 proves the fixed-space lemma used to choose the mixing unitary. Section 4 establishes the planar boundary theorem. Section 5 constructs the bands from the original group relation and removes their connected components from the inside out, proving g=1.

The proof uses group presentations, the spectral theorem for unitary operators, elementary surface topology, and degree modulo two. The trace, incidence-space, and diagram calculations are included.

2 Spectral phase and its trace

We measure a unitary by the sum of its spectral angles, with each angle chosen in [0,1). The value zero at the cut is essential: the resulting quantity is subadditive and vanishes only at the identity.

Let A be an arbitrary group, with identity 1. On \ell^2(A) define the left and right translations by

and let \mathcal M be the commutant of all R(a). Thus L(a)\in\mathcal M. For T\in\mathcal M, set \tau(T)=T_{1,1}, and for a finite matrix (T_{ij})\in M_b(\mathcal M) set

Throughout, the matrix trace is unnormalized: \tau_b(I)=b.

We record why these are faithful traces, including when A is uncountable. Writing t_x=T_{x,1}, commutation with right translations gives T_{x,y}=t_{xy^{-1}}, and (t_x)_{x\in A}\in\ell^2(A). For S,T\in\mathcal M,

Both sums converge absolutely by Cauchy–Schwarz; square-summable families have countable support even if A is uncountable. Positivity is immediate. If T\geq0 and \tau(T)=0, then T^{1/2}\delta_1=0. Since T^{1/2} also commutes with right translations, it kills every basis vector, so T=0. Block multiplication proves traciality of \tau_b; the same square-root argument, applied to each e_j\otimes\delta_1, proves its faithfulness.

The algebra M_b(\mathcal M) is the commutant of the simultaneous right translations on \C^b\otimes\ell^2(A). For background on group von Neumann algebras and functional calculus, see [6, Sections 2.2, 3.3, and 3.4, EP1]. It contains spectral projections and bounded Borel functions of its normal elements. For a unitary on an arbitrary Hilbert space, the spectral calculus can be obtained by decomposing the space into orthogonal cyclic reducing subspaces: each is the closed span of \{U^j\xi:j\in\Z\} and is separable, and the calculi on these subspaces form a direct sum. Uniqueness of the spectral resolution shows that the calculus commutes with every unitary commuting with U, so it remains in the right-translation commutant. Moreover, \tau_b is the finite sum of the vector functionals at e_j\otimes\delta_1. In particular, for a fixed unitary U, uniformly bounded Borel functions f_r converging pointwise to f satisfy

Indeed, apply dominated convergence to the finite scalar spectral measures associated with these vectors. This also explains why no separability assumption is needed.

For \T=\{z\in\C:|z|=1\}, define the Borel function

Thus 0\leq h(U)\leq I and e^{2\pi i h(U)}=U. Faithfulness implies

3 A unitary with a large fixed space

The next lemma supplies the matrix that will mix copies of a word disk. Its conclusion holds for an arbitrary fixed unitary matrix; no continuous choice of the mixing matrix is needed. The compact-unitary degree strategy goes back to Gerstenhaber–Rothaus [3, Theorem 1]; see also [9, Section 1.1] for removing coefficients by homotopy. Here the degree domain records a fixed plane as well as a unitary, so that it has the dimension required for the intersection.

4 A planar surface cannot have one nontrivial boundary

We now combine the two unitary lemmas. Corner-labelled relative diagrams are classical; see Howie [5, Section 1, Lemma 1]. Diagram obstructions and Euler-characteristic estimates also appear in [12, preprint Sections 3–4, especially Lemma 2] and [2, preprint Sections 3–4]. We give the disk-and-band argument in full, using the spectral-phase comparison for its central estimate. Fix a group A and a cyclic word

Thus n is odd and positive. The coefficients may equal 1. Indices on this word are taken modulo n.

A word disk is an oriented disk whose boundary has n disjoint marked intervals, called slots, separated by labelled arcs, called corners. A positive disk reads w around its oriented boundary, and a negative disk reads w^{-1}. Each slot is indexed by the occurrence of t in (6) from which it comes. Its traversal sign is s_i on a positive disk and -s_i on a negative disk. The sign of a disk and the traversal sign of a slot are different data.

Attach rectangular bands to pairs of slots, compatibly with orientation, using every slot exactly once. Require each band to join opposite traversal signs. The resulting compact oriented surface has boundary words in A: read the corner labels in the boundary orientation, ignoring band sides. Changing the starting point conjugates the word, which does not affect whether it represents 1.

4.1 Slots, corners, and boundary cycles

The total of all traversal signs is zero, since the bands pair opposite signs. A positive disk contributes \sum_i s_i and a negative disk its negative. There are therefore k disks of each kind, with k\geq1. Let D be the set of their 2nk slots. Define two permutations of D: \sigma takes each slot to the next slot on its oriented disk, and \theta pairs the two slots of a band. Write c_d\in A for the corner from d to \sigma d. Thus

The boundary circles are the cycles of \sigma\theta. Indeed, from the start of a slot d, a boundary path follows a band side to the finish of \theta d, then follows its corner to the start of \sigma\theta d. For a cycle z=(d_1,\ldots,d_{d_z}) in this order, its word is

Figure 1 records this local convention.

There are 2k disks and nk bands. If q is the number of boundary circles, the Euler characteristic gives

In particular q\geq1: the surface has nonempty boundary along its corners.

We will assign a small real phase \eta_z to each boundary circle, with \sum_z\eta_z=0. The phases will let us approach the spectral cut from opposite sides on the trivial and possibly nontrivial boundary circles. They can be realized by numbers \gamma_d\in\R satisfying

To see this, form a graph whose vertices are the cycles of \sigma\theta and whose edges are the pairs \{d,\theta d\}. The graph is connected: connectedness of \Sigma makes \langle\sigma,\theta\rangle transitive on D, and this group equals \langle\sigma\theta,\theta\rangle. On a spanning tree, any assignment of total sum zero is the divergence of a real edge flow. For completeness, remove a leaf, put its required value on its end of the incident edge and its negative on the other end, and adjust the value required at the adjacent vertex. The final vertex has value zero. Set all other edge flows to zero, including loops. This gives (10); if q=1, take every \gamma_d=0.

4.2 An upper bound obtained by mixing the disks

Use the algebra M_{2nk}(\mathcal M) and its trace from Section 2, with matrix coordinates indexed by D. On \C^D\otimes\ell^2(A) define

Both operators are unitary; (10) gives \Theta^2=I. The product S\Theta follows the boundary permutation \sigma\theta. Its coefficient weights act on the left, so a complete circuit multiplies the inverse corner labels in reverse order, giving the inverse boundary word. We regard scalar matrices on \C^D as elements of this algebra by tensoring with the identity. For a scalar orthogonal projection Q, \tau_{2nk}(Q\otimes I)=\operatorname{rank}Q, since \tau(I)=1.

Number the disks of each kind by 1,\ldots,k. At each slot index i, write e_{+,i}(u) and e_{-,i}(u) for the positive- and negative-disk arrays with coefficient vector u\in\C^k. Given X\in\U(k), put

This scalar operator is a unitary involution. The corner convention gives

The same coefficient weight acts on all k disks of a given kind. Consequently JSJ=S^{-1}: on a positive array both sides shift to i-1 with weight L(a_{i-1}), and on a negative array both shift to i+1 with weight L(a_i^{-1}). Thus SJ is a self-adjoint unitary. It exchanges disk kinds, so its trace is zero. Its negative spectral projection (I-SJ)/2 has trace nk, and hence

We now choose X to make J\Theta have a large fixed space. Let V_+ and V_- be the scalar coordinate spaces of slots with positive and negative traversal signs. Each has dimension nk. Both J and \Theta exchange them. In V_+, the block at index i consists of positive disks when s_i=1 and negative disks when s_i=-1; in V_-, use the opposite kind. Order each block by disk number. In these bases \Theta:V_+\to V_- is a fixed matrix P\in\U(nk), while J:V_-\to V_+ is \diag(X^{-s_1},\ldots,X^{-s_n}). Thus

Lemma 3.1, applied to the signs -s_i, supplies X for which this block has at least k fixed directions. Moreover, J(J\Theta)J=(J\Theta)^{-1}, and J exchanges V_+ and V_-. The other block therefore also has at least k fixed directions.

The eigenvalues of the scalar unitary J\Theta are paired with their inverses, with multiplicities. Since h(\lambda)+h(\lambda^{-1})=1 for \lambda\ne1 and both terms vanish at 1, we obtain

Apply Lemma 2.1 to S\Theta=(SJ)(J\Theta):

This bound holds for every choice of the boundary phases in (10). The choice of X may depend on those phases.

4.3 Computing the phase along boundary circles

We compute the same trace directly from the boundary cycles. On a cycle z=(d_1,\ldots,d_{d_z}), one circuit of S\Theta starting at d_1 has weight

Circuit weights at other starting points are unitary conjugates. If T_z is the restriction of S\Theta to this cycle, then

Here and below \ell uses the matrix trace of the appropriate size. To prove the formula, let d=d_z and \zeta=e^{2\pi i/d}. Rephasing the successive coordinate vectors by 1,\zeta,\ldots,\zeta^{d-1} conjugates T_z to \zeta T_z. For every \lambda\in\T the exact identity

follows by listing the d fractional parts; it also holds when \lambda^d=1, because h(1)=0. Apply bounded Borel functional calculus and take the trace. Each term on the left has trace \ell(T_z). The constant on the right has trace d(d-1)/2, and T_z^d is diagonal with d conjugate circuit weights. Division by d gives (17).

5 From a coefficient relation to a planar surface

We now prove Theorem 1.1. The construction has two steps. A finite product of conjugates of the relator gives a punctured disk with paired bands. We then remove its connected band surfaces from the inside out, applying Theorem 4.1 at each step. The relation-to-surface interpretation is also described in [2, preprint Section 2, especially Example 2.5]. We retain the finite domain and the original coefficient map explicitly in the construction.