A group A and an infinite cyclic group generated by t have free product H = A * <t>. The homomorphism p of H onto Z sends every element of A to 1 and sends t to 1. An element w of H is unimodular when p(w) is 1 or -1. Splitting powers of t into single letters, a unimodular word has the cyclic form
w = t^{s_1} a_1 t^{s_2} a_2 ... t^{s_n} a_n
with a_i in A, s_i in {1,-1}, and sum_i s_i equal to 1 or -1. The number n of letters is then odd. A coefficient a_i may equal the identity.
Normal closure. Let N(w) be the normal closure of w in H. The equation w(t)=1 is solvable over A when some overgroup B contains A and some b in B satisfies w(b)=1. Every such solution factors through H/N(w); conversely an injective coefficient map A -> H/N(w) lets the quotient itself serve as B with b the image of t. Solvability over A is therefore equivalent to injectivity of A -> H/N(w).
Main statement. For every group A and every unimodular w in H, the map A -> H/N(w) is injective.
Kervaire corollary. If A is nontrivial, then H/N(w) is nontrivial for every w in H. Proof. For p(w) equal to 1 or -1 this is the main statement. For p(w)=d different from 1 and -1, compose p with the quotient Z -> Z/dZ, reading Z/0Z as Z; the composite kills w and is onto a nontrivial group, so it factors through a surjection H/N(w) -> Z/dZ.
Place in the literature. Kervaire's theory of higher dimensional knots exhibited groups of weight one, since a knot group is normally generated by a meridian; the question for H/N(w) is whether one relator can normally generate the whole free product. Gerstenhaber and Rothaus solved nonsingular systems of equations, in which the unknown occurs with nonzero exponent sum, in finite overgroups by a compact-unitary argument; that unitary-group device reappears in the fixed-plane step. Pestov surveys hyperlinear groups, the groups embeddable into metric ultraproducts of unitary groups. Klyachko proved the unimodular case for torsion-free groups, and showed that the nontriviality assertion for all relators is equivalent to the statement that adjoining one generator and one relator never turns a nonsimple group into a simple one. Klyachko and Lurye proved solvability after the extra relation w^m=1 for unimodular w and m at least 2. Chen compared the complexity of a surface with the complexity of its boundary in HNN extensions, obtaining a Freiheitssatz that reproves the torsion-free case and generalizes the powered relator theorem. Kawauchi published a confirmation through knot exteriors, relative collapses, and ribbon sphere-link groups, proving Conjecture Z of Gonzalez Acuna and Ramirez, whose endpoint is the nontriviality of H/N(w) for every w.
The abstract of the Kawauchi paper states:
"Kervaire conjecture that the weight of the free product of every
non-trivial group and the infinite cyclic group is not one is
affirmatively confirmed by confirming affirmatively Conjecture Z on
the knot exterior ... a conjecture equivalent to Kervaire conjecture."
./kervaire-1965-noeuds-dimensions-superieures.pdf
./pestov-0804.3968-hyperlinear-sofic-guide.pdf
./klyachko-math0409146-kervaire-laudenbach-simple-groups.pdf
./klyachko-lurye-1010.4220-powered-unimodular-relator.pdf
./chen-2302.09811-minimal-complexity-surfaces.pdf
./kawauchi-2303.13728-ribbon-sphere-link.pdf
The argument adds two independent tools to the surface picture. One is a fixed-plane lemma for unitaries, proved by degree modulo two on a compact manifold that records a unitary together with a fixed k-plane. The other is subadditivity of the trace of the fractional spectral phase of a unitary on a finite matrix amplification of a group von Neumann algebra. Their combination produces a single obstruction for planar surfaces: in a connected genus-zero surface built from letter-blocks and bands, at most one boundary word can fail to be trivial.
A trace and the spectral phase. Let A be any group with identity e. On the Hilbert space l^2(A) with orthonormal basis delta_x (x in A) put
L(a) delta_x = delta_{a x}, R(a) delta_x = delta_{x a^{-1}}.
Let M be the commutant of the maps R(a); then each L(a) lies in M. For T in M set tau(T) = <T delta_e, delta_e>, and for a b by b matrix (T_{ij}) over M set
tau_b((T_{ij})) = sum_j tau(T_{jj}).
This matrix trace is unnormalized: tau_b(I) = b.
Faithfulness. Write t_x = T_{x,e}. Commutation with right translations gives T_{x,y} = t_{x y^{-1}}, and the family (t_x) lies in l^2(A). For S,T in M,
tau(S T) = sum_x s_{x^{-1}} t_x = sum_x t_{x^{-1}} s_x = tau(T S),
both sums converging absolutely by Cauchy-Schwarz, and a square-summable family is supported on a countable set even when A is uncountable. Positivity is immediate. If T >= 0 and tau(T) = 0, then T^{1/2} delta_e = 0; since T^{1/2} also commutes with right translations it kills every basis vector, so T = 0. Block multiplication makes tau_b tracial, and the same square-root argument applied to each e_j (x) delta_e makes tau_b faithful.
Borel calculus. The algebra M_b(M) contains the spectral projections and bounded Borel functions of its normal elements. For a unitary U the Borel calculus is assembled from the separable reducing subspaces spanned by the vectors U^j xi (j in Z); uniqueness of the spectral resolution makes the calculus commute with every unitary commuting with U, so it remains in M_b(M). For a fixed unitary U and uniformly bounded Borel functions f_r tending to f pointwise,
tau_b(f_r(U)) -> tau_b(f(U)),
by dominated convergence on the finitely many scalar spectral measures of the vectors e_j (x) delta_e.
Phase. Let T be the unit circle. Define h on T by h(e^{2 pi i x}) = x - floor(x) for x in [0,1), with h(1) = 0, and set ell(U) = tau_b(h(U)). Then 0 <= h(U) <= I and e^{2 pi i h(U)} = U. Faithfulness gives ell(U) = 0 exactly when U = I.
Subadditivity. For all unitaries U, V in M_b(M),
ell(U V) <= ell(U) + ell(V).
Proof. Put B = h(V) and W_u = U e^{2 pi i u B} for 0 <= u <= 1, so W_0 = U, W_1 = U V, and W'_u = 2 pi i W_u B. For smooth f on T write f' for the derivative in the angle variable, f'(e^{2 pi i x}) = d/dx f(e^{2 pi i x}). The identity
d/du tau_b(f(W_u)) = tau_b(f'(W_u) B)
holds. For f(z) = z^r with r > 0 the product rule and traciality give
d/du tau_b(W_u^r) = 2 pi i sum_{j=0}^{r-1} tau_b(W_u^{j+1} B W_u^{r-1-j}) = 2 pi i r tau_b(W_u^r B),
negative r follows from (W_u^{-1})' = - 2 pi i B W_u^{-1}, and r = 0 is immediate. A smooth f has Fourier coefficients with sum_r (1+|r|) |a_r| finite, so the Fourier and differentiated Fourier series converge uniformly and the differentiated traces are bounded termwise by 2 pi b |r a_r| |B|; termwise differentiation gives the identity.
Now take f real with f' <= 1 everywhere. Since B >= 0 and f' is real-valued, f'(W_u) is Hermitian with spectrum at most 1, hence f'(W_u) <= I; by traciality and positivity of B,
tau_b(f'(W_u) B) = tau_b(B^{1/2} f'(W_u) B^{1/2}) <= tau_b(B).
Integration over u gives
tau_b(f(U V)) <= tau_b(f(U)) + tau_b(h(V)).
To reach h, choose a smooth nonnegative density rho_eps of integral one supported in (0,eps) with 0 < eps < 1, and set
f_eps(e^{2 pi i x}) = integral_0^eps rho_eps(s) (x + s - floor(x + s)) ds.
These f_eps are smooth and lie in [0,1]. For the periodic fractional part g(x) = x - floor(x), one has g(x+d) - g(x) = d - (floor(x+d) - floor(x)) <= d for d >= 0; averaging and differentiating gives f_eps' <= 1. Also f_eps tends to h pointwise, and at the single discontinuity
0 <= f_eps(1) = integral_0^eps s rho_eps(s) ds <= eps,
so the limiting value at 1 is h(1) = 0. Applying the smooth inequality to f_eps and passing to the limit with the bounded-convergence property for the fixed unitaries U and U V gives the claim.
Thompson's exponential formula gives the same inequality for finite matrices directly: a product of unitaries equals e^{2 pi i C} with C a sum of unitary conjugates of h(U) and h(V), so C >= 0 and h(e^{2 pi i C}) <= C yields the bound after taking traces. The smoothing route keeps the prescribed value at the phase cut and needs no separability or approximation hypothesis.
A fixed-plane lemma. Let s_1,...,s_n lie in {1,-1} with eps = sum_i s_i in {1,-1}, let k >= 1 be an integer, and let m = nk. For every P in U(m) there is X in U(k) such that
F(X) = diag(X^{s_1}, ..., X^{s_n}) P
fixes a complex subspace of dimension at least k.
Proof. The plan is a degree modulo two count on a compact manifold that records a unitary together with a fixed k-plane.
Homotopy to a normal form. Move P to I_m along a smooth path in U(m) obtained by diagonalizing P and interpolating its angles; each subsequent homotopy can be reparametrized to be constant near the endpoints. On two coordinate k-planes set
R_u = [[cos(pi u/2) I_k, -sin(pi u/2) I_k],
[sin(pi u/2) I_k, cos(pi u/2) I_k]], 0 <= u <= 1.
Then diag(X, I_k) R_u diag(I_k, X^{-1}) R_u^{-1} is a smooth function of (u,X) running from diag(X, X^{-1}) at u=0 to I_{2k} at u=1. Apply this on each pair consisting of one block of sign +1 and one of sign -1; the leftover block has sign eps. After also moving the constant identity blocks to -I, the map F is homotopic, through maps of the same form with X in U(k), to
F_0(X) = X^eps (+) (-I_r), r = m - k.
Incidence manifold. Let Gr_k(C^m) be the Grassmannian of complex k-planes and set
R = { (W,E) in U(m) x Gr_k(C^m) : W acts as the identity on E }.
A unitary fixing E preserves E^perp, where it is an arbitrary unitary of dimension r, so R is a smooth bundle over Gr_k(C^m) with fiber U(r); local orthonormal frames of E^perp give product charts. The closed condition (W - I_m) p_E = 0, with p_E the orthogonal projection onto E, places R inside a compact product. Base and fiber are connected, so R is a compact connected boundaryless manifold of real dimension
2kr + r^2 = m^2 - k^2.
Hence U(k) x R and U(m) are connected closed manifolds of the same dimension m^2.
Map and unique preimage. Consider
G_0 : U(k) x R -> U(m), G_0(X,W,E) = F_0(X) W^{-1}.
The only preimage of I_m is (X,W,E) = (I_k, W_0, E_0), where W_0 = I_k (+) (-I_r) and E_0 = C^k (+){0}. Indeed G_0 = I_m gives W = F_0(X); the fixed space of F_0(X) = X^eps (+) (-I_r) is Fix(X^eps) (+){0}, so a fixed k-plane E must equal E_0 and Fix(X) must have dimension k, forcing X = I_k and then W = W_0.
Regularity. Let u(j) denote the real space of skew-Hermitian j by j matrices. Tangent coordinates at the preimage are A in u(k), B in u(r), Z in M_{r x k}(C). Put K_Z = [[0, -Z*],[Z, 0]] and take the charts
X = e^A, E = e^{K_Z} E_0, W = e^{K_Z} diag(I_k, -e^B) e^{-K_Z}.
The block Z gives the Grassmannian tangent directions and B gives the fiber directions. At zero,
F'_0 = diag(eps A, 0), W' = [K_Z, W_0] + diag(0, -B) = [[0, 2Z*],[2Z, -B]].
Since F_0 = W = W_0 = W_0^{-1} at the preimage,
dG_0(A,B,Z) = F'_0 W_0 - W' W_0 = [[eps A, 2Z*],[-2Z, -B]].
Every skew-Hermitian m by m matrix has this form uniquely, so dG_0 is an isomorphism and I_m is a regular value. When r = 0 the formulas reduce to A -> eps A, and the same formulas also cover k = 1.
Degree conclusion. For smooth maps between connected closed manifolds of one dimension, the parity of a regular fiber is independent of the regular value and invariant under smooth homotopy; this is the degree modulo two, for which no orientations are needed; Milnor's book on the differentiable viewpoint is the standard reference. The regular preimage (I_k, W_0, E_0) gives deg_2(G_0) = 1. The homotopy from F_0 to F induces a homotopy from G_0 to G(X,W,E) = F(X) W^{-1}, so deg_2(G) = 1 as well. A map of nonzero degree onto a connected closed manifold is surjective, so I_m is attained: some X in U(k) and some (W,E) in R satisfy F(X) = W, and W fixes E, a plane of dimension k. This proves the lemma.
./milnor-1965-differentiable-viewpoint.pdf
Planar surfaces. Fix a group A and a cyclic word
w = t^{s_1} a_1 t^{s_2} a_2 ... t^{s_n} a_n, a_i in A, s_i in {1,-1}, sum_i s_i = 1 or -1,
so n is odd. A word disk is an oriented disk whose boundary carries n disjoint marked intervals, the slots, separated by labelled arcs, the corners. A positive disk reads w around its oriented boundary; a negative disk reads w^{-1}. Each slot is indexed by its t-letter, and 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, and require every band to join opposite traversal signs. The result is a compact oriented surface whose boundary reads words in A: corner labels are read along the boundary orientation, with band sides ignored. Moving the starting point conjugates the word and does not change whether it equals 1.
Boundary theorem. Let Sigma be a connected genus-zero surface built from a nonempty finite collection of word disks for w and bands that pair opposite traversal signs. If every boundary word except possibly one is trivial in A, then the remaining boundary word is trivial as well.
Proof. The total traversal sign is zero because bands pair opposite signs, so positive disks contribute sum_i s_i and negative disks its negative; hence there are k positive and k negative disks, with k >= 1. Let D be the set of the 2nk slots. Two permutations act on D: sigma sends a 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. Then
positive disk: i -> i+1 with corner a_i, negative disk: i+1 -> i with corner a_i^{-1}.
The boundary circles are the cycles of sigma theta: from the start of d a boundary path crosses the band to the finish of theta d, then follows the corner to the start of sigma theta d. For a cycle z = (d_1,...,d_{d_z}) the boundary word is
r_z = c_{theta d_1} c_{theta d_2} ... c_{theta d_{d_z}}.
The surface has 2k disks and nk bands, so its Euler characteristic is chi = 2k - nk = 2 - q, where q is the number of boundary circles; thus q = nk - 2k + 2 >= 1.
Phases. Assign a real phase eta_z to each boundary circle with sum_z eta_z = 0. These phases are realized by real numbers gamma_d with
gamma_{theta d} = - gamma_d, sum_{d in z} gamma_d = eta_z.
To see this, form a graph whose vertices are the cycles of sigma theta and whose edges are the pairs {d, theta d}. Connectivity of Sigma makes the group generated by sigma and theta transitive on D, and that group equals the group generated by sigma theta and theta, so the graph is connected. On a spanning tree any assignment of total sum zero is the divergence of an edge flow: remove a leaf, put its value on the incident edge with opposite sign on the other end, and adjust the adjacent vertex; the last vertex has value zero. Setting all other edge flows to zero, including loops, gives the required gamma_d, and for q = 1 take every gamma_d = 0.
Upper bound by mixing. Work on C^D (x) l^2(A) inside the algebra M_{2nk}(M) with the unnormalized trace tau_{2nk} attached to A, matrix coordinates indexed by D. Define
S(e_d (x) v) = e_{sigma d} (x) L(c_d^{-1}) v, Theta(e_d (x) v) = e^{2 pi i gamma_d} e_{theta d} (x) v.
Both are unitary, and Theta^2 = I by the phase relations. The product STheta follows the permutation sigma theta; its coefficient weights act on the left, and a full circuit multiplies the inverse corner labels in reverse order, giving the inverse boundary word.
Number the disks of each kind by 1,...,k. At slot index i let e_{+,i}(u) and e_{-,i}(u) be the positive- and negative-disk arrays with coefficient vector u in C^k. For X in U(k) put
J e_{+,i}(u) = e_{-,i}(X u), J e_{-,i}(u) = e_{+,i}(X^{-1} u).
This scalar operator is a unitary involution. The corner convention gives
S e_{+,i}(u) = e_{+,i+1}(u) (x) L(a_i^{-1}), S e_{-,i}(u) = e_{-,i-1}(u) (x) L(a_{i-1}),
and the same coefficient weight acts on all k disks of a given kind, so JSJ = S^{-1}. Hence SJ is a self-adjoint unitary exchanging the two disk kinds, with trace zero; the projection (I - SJ)/2 has trace nk, and ell(SJ) = nk/2.
Let V_+ and V_- be the scalar coordinate spaces of slots with positive and negative traversal sign, each of dimension nk. Both J and Theta exchange the two spaces. In V_+, the block at index i is a positive disk when s_i = 1 and a negative disk when s_i = -1; in V_-, the opposite kind. In these bases Theta: V_+ -> V_- is a fixed unitary P, while J: V_- -> V_+ is diag(X^{-s_1},...,X^{-s_n}); hence (JTheta)|_{V_+} = diag(X^{-s_1},...,X^{-s_n}) P. The fixed-plane lemma applied to the signs -s_i (whose sum is again 1 or -1) yields X for which this block has at least k fixed directions. Since J(JTheta)J = (JTheta)^{-1} and J exchanges V_+ and V_-, the other block also has at least k fixed directions. The eigenvalues of the scalar unitary JTheta are paired with inverse eigenvalues, and h(lambda) + h(lambda^{-1}) = 1 for lambda different from 1, with both terms zero at 1, so
ell(JTheta) = (2nk - dim Fix(JTheta))/2 <= nk - k.
Subadditivity for STheta = (SJ)(JTheta) gives
ell(STheta) <= 3nk/2 - k.
This bound holds for every choice of the boundary phases.
Phase along the boundary circles. The same trace is computed cycle by cycle. On a cycle z = (d_1,...,d_{d_z}) one circuit of STheta starting at d_1 has weight
e^{2 pi i eta_z} L(c_{theta d_{d_z}}^{-1}) ... L(c_{theta d_1}^{-1}) = e^{2 pi i eta_z} L(r_z^{-1}),
and circuit weights at other starting points are unitary conjugates. If T_z is the restriction of STheta to this cycle, then
ell(T_z) = (d_z - 1)/2 + tau(h(e^{2 pi i eta_z} L(r_z^{-1}))).
For the proof let d = d_z and zeta = e^{2 pi i/d}. Listing of fractional parts gives, for lambda on the unit circle,
sum_{j=0}^{d-1} h(zeta^j lambda) = (d-1)/2 + h(lambda^d),
valid also when lambda^d = 1 because h(1) = 0. Diagonal rephasing of the d coordinate slots conjugates T_z to zeta T_z; applying the functional calculus to T_z and using traciality and unitary invariance of tau turns the identity into the displayed formula.
Proof of the boundary theorem. Let z_0 be the possibly nontrivial boundary circle. Approach the phase cut from below on the trivial circles and from above on z_0: for eps > 0 put eta_z = -eps for z different from z_0 and eta_{z0} = (q-1) eps, so the phases sum to zero. On each trivial circle the last term of the cycle formula tends to 1 because h(e^{2 pi i eta_z}) tends to 1 from below, while on z_0 it tends to tau(h(L(r_{z0}^{-1}))) from the nonnegative side. Since sum_z d_z = 2nk and q = nk - 2k + 2,
lim ell(STheta) = (2nk - q)/2 + (q - 1) + tau(h(L(r_{z0}^{-1}))) = 3nk/2 - k + tau(h(L(r_{z0}^{-1}))).
The uniform bound forces tau(h(L(r_{z0}^{-1}))) = 0; faithfulness of tau and positivity of h give h(L(r_{z0}^{-1})) = 0, hence L(r_{z0}^{-1}) = I and r_{z0} = 1. For q = 1 the phase of the single circle is zero and the same conclusion holds directly.
From a coefficient relation to a planar surface. The main statement follows from a planar surface built from the relator. A based CW space Y with pi_1(Y) = A exists for every group: take one loop x_a per element of A and attach two-cells imposing x_1 = 1 and x_a x_b = x_{ab}, so that pi_1(Y) = A. Then pi_1(Y vee T) = A * <t>, with the added circle representing t. Let Delta be a polygonal disk with the interiors of r disjoint polygonal disks removed, giving a planar surface P. Suppose g in A lies in the normal closure of w, so for some finite r
g = product_{j=1}^{r} b_j w^{eps_j} b_j^{-1}, b_j in H, eps_j in {1,-1}.
If r = 0 then g = 1; assume r > 0.
./hatcher-2002-algebraic-topology.pdf
There is a continuous map f: P -> Y vee T whose outer boundary is a based loop in Y representing g and whose j-th inner boundary reads w^{eps_j} when oriented as the boundary of its missing disk. Join marked points on the inner circles to a common outer basepoint by otherwise disjoint arcs; order the arcs so the outer loop represents the ordered product of the conjugated inner loops, and map the j-th arc to a path representing b_j. Use based coefficient loops in Y and one monotone traversal of T, affine in angle, for each letter t or t^{-1}; on an inverse relator reverse the chosen paths in reverse order. Cutting along the arcs gives a disk whose boundary path is nullhomotopic by the displayed product; extend over that disk and reglue the matching arc copies. This uses equality in pi_1(Y vee T) before imposing w.
./chen-2302.09811-minimal-complexity-surfaces.pdf
Write u: P -> T for the projection collapsing Y and v: P -> Y for the projection collapsing T. The map u is constant on the outer circle and on every coefficient interval, and traverses T once on each t-letter interval.
Extracting bands. Identify T with R/Z with wedge point 0. Approximate u, relative to the boundary, by a map u' that is piecewise affine in local angle coordinates, with uniform circle distance below 1/12: subdivide the boundary so its angle maps are affine on edges, choose a fine finite triangulation of P, lift angles on each triangle, interpolate the vertex values affinely, and note that lifts on shared edges differ by an integer so the circle maps agree there and equal u on boundary edges. Only the compact surface P and its circle map are triangulated.
Choose an angle z in (1/3,2/3) outside the images of all vertices. The level u'^{-1}(z) is a finite disjoint union of polygonal circles and properly embedded polygonal arcs: inside each triangle it is a segment or empty, and it misses every vertex. Since u' = u on the boundary, each t-letter interval contains exactly one arc endpoint, and there are no others. Ignore the closed level circles.
The circle orientation coorients the arcs, and the arc endpoints carry opposite crossing signs along the induced boundary orientation of P. The missing-disk orientation reverses that orientation at both ends, so each arc pairs opposite t-letter signs even when both ends lie on one disk.
At every point of the level the original value u has distance above 1/4 from the wedge point, so f maps an open neighborhood of the level into T minus {0}, where v is constantly the basepoint. Disjoint thin rectangular neighborhoods of the arcs inside this neighborhood give bands meeting the missing-disk boundaries in small intervals inside the t-letter intervals; v is constant on every band.
Each attachment interval is a slot with the sign of its original t-letter. Between consecutive slots the projected boundary path is the original coefficient loop together with constant pieces from adjacent t-intervals, so its label in A is unchanged: positive disks have corners a_i and negative disks a_i^{-1}. The geometric slots may therefore be smaller than the original letter intervals.
Removing the planar components. Put the missing disks back into Delta as word disks and adjoin all the bands. The union S is a finite compact planar surface, possibly disconnected. Each component contains a word disk, every slot is used once, the paired signs are opposite, and the orientation induced from the plane satisfies the hypotheses of the boundary theorem; along its boundary v reads exactly the corner words of that theorem, because band sides and remaining pieces of t-intervals are constant in Y.
The nonexterior boundary words of a component need not be trivial, since other components may lie in the regions bounded by those circles. Choose an innermost component: a connected planar component C has a unique exterior boundary circle, whose closed Jordan disk is D(C), and its other boundary circles bound inner complementary disks. Among the finitely many D(C) choose one minimal under inclusion. Then no inner complementary disk of this C contains another component, since such a component would have its exterior disk strictly inside D(C), contradicting minimality.
Each inner complementary disk of C contains no word disks. It already carries the map v to Y, so its boundary word is trivial in A. The boundary theorem makes the exterior boundary word of C trivial as well, so the actual boundary loop of D(C) under v is nullhomotopic in Y. Fill D(C) by a map to Y agreeing with that loop, and discard the word disks and bands of C.
Write v for the resulting coefficient map. The replacement changes nothing outside D(C) and affects no other component. It leaves a continuous Y-map outside the remaining word-disk interiors, with the same corner readings and the same constant remaining bands, which is the invariant the argument needs to repeat. Each step removes one component, so after finitely many steps the original outer loop extends over Delta in Y. Therefore g = 1 in A, and the map A -> H/N(w) is injective.
This proves the main statement. For a word with p(w) different from 1 and -1, composing p with Z -> Z/dZ kills w and is onto a nontrivial group, so H/N(w) is nontrivial; for p(w) equal to 1 or -1 the main statement applies directly.