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The minus-one specialization of three-strand Burau has kernel generated by Delta to the fourth

Statement

Assume AC (inherited through the identification of the reduced matrices with the topological representation). Let $Δ = σ_{1} σ_{2} σ_{1}$ be the half twist of $B_{3}$ and let ${\overset{ˉ}{ρ}}_{3} : B_{3} \to {GL}_{2} (Λ_{1})$ be the reduced Burau representation. Evaluate the matrices of The topological and matrix Burau representations agree at $t = - 1$ and call the resulting homomorphism ${\overset{ˉ}{ρ}}_{3} (- 1) : B_{3} \to {GL}_{2} (Z)$ ; in the basis $(g_{1}, g_{2})$ it sends $σ_{1} ⟼ U = (\begin{array}{cc} 1 & - 1 \\ 0 & 1 \end{array}), σ_{2} ⟼ V = (\begin{array}{cc} 1 & 0 \\ 1 & 1 \end{array}) .$ Then $U, V$ generate ${SL}_{2} (Z)$ ; the group with presentation $⟨ U, V ∣ U V U = V U V, (U V U)^{4} = 1 ⟩$ is isomorphic to ${SL}_{2} (Z)$ ; the map ${\overset{ˉ}{ρ}}_{3} (- 1)$ is surjective with image ${SL}_{2} (Z)$ ; and its kernel is $\ker {\overset{ˉ}{ρ}}_{3} (- 1) = ⟨ Δ^{4} ⟩ = {Δ^{4 k} : k \in Z},$ the infinite cyclic central subgroup generated by $Δ^{4}$ . Equivalently, the only braid-group relation added to the Artin presentation of $B_{3}$ by the specialization is $Δ^{4} = 1$ .

Facts & Assumptions

Given: AC; the reduced representation ${\overset{ˉ}{ρ}}_{3}$ in the basis $(g_{1}, g_{2})$ ; the entrywise evaluation $Λ_{1} \to Z$ , $t \mapsto - 1$ ; the matrices $U, V$ above; the braid group $B_{3} = ⟨ σ_{1}, σ_{2} ∣ σ_{1} σ_{2} σ_{1} = σ_{2} σ_{1} σ_{2} ⟩$ ; the half twist $Δ = σ_{1} σ_{2} σ_{1}$ .

[F1]

In the basis $(g_{1}, g_{2})$ , ${\overset{ˉ}{ρ}}_{3} (σ_{1}) = (\begin{array}{cc} - t & t \\ 0 & 1 \end{array})$ and ${\overset{ˉ}{ρ}}_{3} (σ_{2}) = (\begin{array}{cc} 1 & 0 \\ 1 & - t \end{array})$ ; a homomorphism into ${GL}_{2} (Λ_{1})$ composes with the ring homomorphism $Λ_{1} \to Z$ , $t \mapsto - 1$ , to give a homomorphism into ${GL}_{2} (Z)$ (The topological and matrix Burau representations agree, The reduced Burau representation, The Laurent polynomial ring as the principal localisation of Z[t] at t).

[F2]

Put $S = (\begin{array}{cc} 0 & - 1 \\ 1 & 0 \end{array})$ and $T = (\begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array})$ . By definition, the quotient $q : {SL}_{2} (Z) \to {PSL}_{2} (Z)$ has kernel ${\pm I} = ⟨ S^{2} ⟩$ .

[F3]

Let $C_{4} = ⟨ a ∣ a^{4} = 1 ⟩$ , $C_{6} = ⟨ b ∣ b^{6} = 1 ⟩$ and let $C_{4} *_{C_{2}} C_{6}$ be the amalgam identifying the order-two subgroup $⟨ a^{2} ⟩$ with the order-two subgroup $⟨ b^{3} ⟩$ . Then $C_{4} *_{C_{2}} C_{6}$ has the presentation $⟨ a, b ∣ a^{4} = 1, b^{6} = 1, a^{2} = b^{3} ⟩$ : applying A free product with amalgamation has the factor presentations plus the amalgamating relations to these two presentations with the amalgamating words $u_{a^{2}} = a^{2}$ , $v_{a^{2}} = b^{3}$ for the generator $a^{2}$ of the common subgroup $C_{2}$ adds exactly the relation $a^{2} b^{- 3} = 1$ , and amalgamated products are pushouts along monomorphisms (Free products with amalgamation along monomorphisms).

[F4]

${PSL}_{2} (Z)$ is the free product $C_{2} * C_{3}$ : with $x$ the class of $S$ and $y$ the class of $S T$ , every element has a unique expression $y^{i_{0}} x y^{i_{1}} x \dots y^{i_{n - 1}} x y^{i_{n}}$ with $i_{j}$ taken modulo $3$ and all inner exponents nonzero, and the emptiness of further relations is proved by the entry-sum argument for products of $S R$ and $S R^{2}$ (Keith Conrad, SL_2(Z), Appendix C, Theorem C.1 and its proof; see the cited locator). In particular the assignment $x \mapsto s$ , $y \mapsto t$ for arbitrary elements $s, t$ of a group with $s^{2} = t^{3} = 1$ extends to a homomorphism $C_{2} * C_{3} \to ⟨ s, t ⟩$ (A free product has the union presentation of presentations of its factors); uniqueness of the normal form is the cited external theorem.

[F5]

The center of $B_{n}$ for $n \geq 3$ is infinite cyclic and generated by the full twist $Δ^{2}$ ; hence $Δ^{4} = (Δ^{2})^{2}$ is central of infinite order (The center of b n is generated by the full twist for n greater than two, The Garside half twist and simple positive braids).

[F6]

Matrix arithmetic is entrywise. For $2 \times 2$ matrices $A, B$ , expanding the four entries of $A B$ in $\det (A B)$ and cancelling cross terms gives $\det (A B) = \det (A) \det (B)$ ; the relevant matrices have determinant $1$ , so their products lie in ${SL}_{2} (Z)$ (Invertible square matrices and similarity over a commutative ring).

[F7]

The matrices $S, T$ generate ${SL}_{2} (Z)$ (Keith Conrad, SL2(Z), Theorem 1.1 and its algebraic proof in Section 2, printed p. 1). That proof applies integer division to the first column, decreasing the absolute value of its lower entry until it vanishes; the resulting upper triangular determinant-one matrix is a signed power of $T$ .

Proof

technique · direct

1.1F1F6algebra

The specialization. Evaluating the matrices of [F1] at $t = - 1$ gives ${\overset{ˉ}{ρ}}_{3} (- 1) (σ_{1}) = U$ and ${\overset{ˉ}{ρ}}_{3} (- 1) (σ_{2}) = V$ as displayed. Computations give $U V U = (\begin{array}{cc} 0 & - 1 \\ 1 & 0 \end{array}) = : S = V U V$ and $S^{4} = I$ (indeed $S^{2} = - I$ ), so the relations $U V U = V U V$ and $(U V U)^{4} = 1$ hold in ${SL}_{2} (Z)$ ; also $\det U = \det V = 1$ , so ${\overset{ˉ}{ρ}}_{3} (- 1)$ takes values in ${SL}_{2} (Z)$ .

1.2F3algebra

The presented group is the amalgam $C_{4} *_{C_{2}} C_{6}$ . Let $P = ⟨ x, y ∣ x y x = y x y, (x y x)^{4} = 1 ⟩$ and set $a = x y x$ , $b = x y$ as words in $P$ . Symbolically, $b^{- 1} a = (x y)^{- 1} (x y x) = y^{- 1} x^{- 1} x y x = y^{- 1} y x = x$ and $a^{- 1} b^{2} = (x y x)^{- 1} (x y)^{2} = x^{- 1} y^{- 1} x^{- 1} x y x y = x^{- 1} y^{- 1} y x y = y$ , so $P = ⟨ a, b ⟩$ ; moreover $(x y)^{3} = (x y x) (y x y) = (x y x)^{2} = a^{2}$ using the relation, so $b^{3} = a^{2}$ , and hence $b^{6} = a^{4} = 1$ . These relations define a homomorphism $θ : C_{4} *_{C_{2}} C_{6} \to P$ sending the amalgam generators to the words $a, b$ . Conversely, in $C_{4} *_{C_{2}} C_{6}$ the elements $x : = b^{- 1} a$ and $y : = a^{- 1} b^{2}$ satisfy $x y x = a$ and $y x y = a$ , so they satisfy the relators of $P$ and define a homomorphism $η : P \to C_{4} *_{C_{2}} C_{6}$ . Direct substitution shows that $η θ$ fixes $a, b$ and $θ η$ fixes $x, y$ , hence the homomorphisms are inverse and $P ≅ C_{4} *_{C_{2}} C_{6}$ .

1.3F2F6F7algebra

Generation. The matrices $S, T$ generate ${SL}_{2} (Z)$ by [F7]. Direct calculation gives $U = T^{- 1}$ and $U V U = S$ , so $⟨ U, V ⟩$ contains both $S$ and $T$ . Since $U, V \in {SL}_{2} (Z)$ by [F6], it follows that $⟨ U, V ⟩ = {SL}_{2} (Z)$ .

2.1F2F3F4step 1.1step 1.3step 1.2

The amalgam is ${SL}_{2} (Z)$ . By von Dyck applied to [F3] with $a \mapsto S$ , $b \mapsto S T$ (which satisfy $S^{4} = I$ , $(S T)^{6} = I$ , $S^{2} = (S T)^{3} = - I$ ), there is a homomorphism $φ : C_{4} *_{C_{2}} C_{6} \to {SL}_{2} (Z)$ , surjective because its image contains $S$ and $S T$ , which generate ${SL}_{2} (Z)$ : indeed $T = S^{- 1} (S T)$ . Let $q : {SL}_{2} (Z) \to {PSL}_{2} (Z)$ be the quotient of [F2]. The composite $ψ = q \circ φ$ sends $a \mapsto \overset{ˉ}{S}$ and $b \mapsto \overline{S T}$ , so it factors through the quotient by $a^{2} = b^{3}$ ; by [F4] its induced map on that quotient is the isomorphism $C_{2} * C_{3} ≅ {PSL}_{2} (Z)$ with generators $a \mapsto \overset{ˉ}{S}$ , $b \mapsto \overline{S T}$ . Its kernel is the normal closure of $a^{2}$ , since quotienting [F3] by $a^{2} = b^{3}$ gives $⟨ a, b ∣ a^{2} = 1, b^{3} = 1 ⟩ = C_{2} * C_{3}$ . The element $a^{2} = b^{3}$ is central in the amalgam, because it is central in both cyclic factors, and has order at most two; its image under $φ$ is $S^{2} = - I \neq I$ , so the kernel of $ψ$ is exactly ${1, a^{2}}$ . If $φ (w) = I$ , then $w \in {1, a^{2}}$ , and the nontriviality of $φ (a^{2})$ forces $w = 1$ . Hence $φ$ is injective and $C_{4} *_{C_{2}} C_{6} ≅ {SL}_{2} (Z)$ . By step 1.2, $P ≅ {SL}_{2} (Z)$ via $x \mapsto U$ , $y \mapsto V$ ; in particular ${\overset{ˉ}{ρ}}_{3} (- 1)$ is surjective onto ${SL}_{2} (Z)$ by steps 1.1 and 1.3.

3.1F5step 2.1∎

The kernel. The Artin presentation of $B_{3}$ has the single relation $σ_{1} σ_{2} σ_{1} = σ_{2} σ_{1} σ_{2}$ , and step 2.1 exhibits ${SL}_{2} (Z)$ as the quotient of $B_{3}$ by the normal closure of $Δ^{4} = (σ_{1} σ_{2} σ_{1})^{4}$ under $σ_{1} \mapsto U$ , $σ_{2} \mapsto V$ . Hence $\ker {\overset{ˉ}{ρ}}_{3} (- 1) = ⟨ ⟨ Δ^{4} ⟩ ⟩$ . By [F5] the element $Δ^{4}$ is central of infinite order, so its normal closure is the cyclic subgroup $⟨ Δ^{4} ⟩ = {Δ^{4 k} : k \in Z} ≅ Z$ . This is the assertion, including the equivalent description as the only relation added to the Artin presentation. AC is inherited through the cited representation agreement; the matrix, presentation and centre computations are choice free.

Depends on

Used by

The reduced Burau representation is faithful for at most three strands Theorem

Dependency tree · two levels

72 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Keith Conrad, SL_2(Z) (author-hosted expository notes), Theorem 1.1 with its Section 2 algebraic proof (printed p. 1), and Appendix C, Theorem C.1 with proof (printed pp. 17-19) (standard reference, not scraped)
Vasudha Bharathram, Joan S. Birman and Tara E. Brendle, The Burau representation is faithful for n = 4, arXiv:2607.05283v1 (6 July 2026), Introduction and section 2 (printed pp. 1-5), and section 4 (Theorem 4.1) (standard reference, not scraped)
Joan S. Birman and Tara E. Brendle, Braids: A Survey (background on Burau matrices, the cyclic cover and absolute homology) (standard reference, not scraped)