Pure braid groups are RFRS

Abstract: Agol in his 2014 ICM proceedings article \cite[Question 11]{Agol14} asks whether braid groups are (virtually) RFRS. We answer this positively by showing that pure braid groups are RFRS. As a consequence, several families of Artin groups are virtually RFRS, including those of type $A_n$ (the braid groups), $B_n=C_n$, $\widetilde A_n$, and $\widetilde C_n$. Our results also provide evidence toward the problem of whether braid groups, and more generally Artin groups, are virtually special; see \cite[Problem 9.4]{HagWi10}, \cite[Problem 13.4]{Wise14}.

• The paper proves that every pure braid group PB_n is RFRS—and, more strongly, RFRp for every prime p—by combining partial towers, planar covers, and semidirect-product stability.
• The proof repeatedly unwraps immersed curves through arc-cyclic covers of planar surfaces, reducing self-intersections and controlling boundary actions so group elements become homologically detectable.
• The result extends to several Artin-group families, including types A, B=C, I_2, affine A, and affine C, providing evidence for virtual specialness while leaving the general conjecture open.

Overview

The paper proves that pure braid groups $PB_n$ are $RFRS$, answering a question posed by Agol in his 2014 ICM proceedings article, in the modified form suggested by Wu–Ye: while braid groups themselves cannot be $RFRS$ (their commutator subgroups are finitely generated and perfect for $n \geq 5$, whereas $RFRS$ groups are locally indicable), the pure braid groups satisfy the property outright. In fact, the stronger statement holds: $PB_n$ is $\mathrm{RFR}p$ for every prime $p$. As consequences, several families of Artin groups — types $A_n$, $B_n = C_n$, $RFRS$0, $RFRS$1, and $RFRS$2 — are virtually $RFRS$3, providing evidence toward the Haglund–Wise problem of whether all Artin groups are virtually special.

The proof strategy combines three ingredients: a detection criterion for $RFRS$4/$RFRS$5 via partial towers, a purely topological "unwrapping" theorem for curves on planar surfaces under arc-cyclic covers, and a boundary control argument ensuring compatibility of isotopical actions with such covers. These assemble into a semidirect-product stability result applied inductively along the Fadell–Neuwirth splitting $RFRS$6.

Partial towers as a detection mechanism

The authors work with the standard characterization of $RFRS$7: a finitely generated group $RFRS$8 is $RFRS$9 if and only if every nontrivial element is detected by a finite partial tower $RFRS$0 of finite-index normal subgroups with $RFRS$1, where detection means the element survives to $RFRS$2 with nonzero rational homology image. They extend this framework to Koberda–Suciu's $RFRS$3 variant, where quotients $RFRS$4 are elementary abelian $RFRS$5-groups. A subtlety they isolate is that the detection condition differs between the two settings: for $RFRS$6, an element is detected when it is excluded by an elementary abelian $RFRS$7-quotient (i.e., lies in $RFRS$8), not when it survives with nontrivial mod-$RFRS$9 homology. The example $n \geq 5$0 with tower $n \geq 5$1 shows that $n \geq 5$2 survives rationally but is invisible to any elementary $n \geq 5$3-quotient of $n \geq 5$4. This distinction forces separate handling of the two cases throughout the paper.

Topologically, the $n \geq 5$5 criterion translates into towers of $n \geq 5$6-covers along which a given element lifts closed and ends up homologically nontrivial. Since surface groups embed in right-angled Artin groups, which are $n \geq 5$7, arbitrary compact surfaces admit such towers; the contribution here is to realize them within the restricted class of arc-cyclic covers of planar surfaces.

Unwrapping curves on planar surfaces

The geometric core is a finite-cover unwrapping theorem. For a minimal immersed representative $n \geq 5$8 of a nontrivial free homotopy class on a compact oriented planar surface, two elementary lemmas supply the setup: an innermost argument produces a simple essential self-intersection subloop $n \geq 5$9, and essential simple closed curves on planar surfaces separate boundary components, so there is a properly embedded arc $RFRS$0 crossing $RFRS$1 exactly once, giving $RFRS$2 for the associated intersection character $RFRS$3.

If $RFRS$4 but $RFRS$5, then $RFRS$6 lifts closed to the arc-cyclic $RFRS$7-cover $RFRS$8, and since the subloop $RFRS$9 changes sheets, that self-intersection is destroyed. Lifting never creates new self-intersections, so the self-intersection count strictly decreases. Crucially, arc-cyclic covers of planar surfaces remain planar: a boundary-counting computation gives $PB_n$0 and Euler characteristic $PB_n$1, forcing genus zero.

Iterating yields the planar tower theorem: for any nontrivial $PB_n$2 and any $PB_n$3, there is a finite tower of arc-cyclic $PB_n$4-covers through which $PB_n$5 lifts closed, terminating at a lift with nonzero integral homology class. Termination follows because the minimal self-intersection number strictly decreases at each step. When the terminal class happens to be divisible by $PB_n$6, a second phase reduces maximal $PB_n$7-divisibility: choosing an arc whose mod-$PB_n$8 character does not vanish on the primitive part $PB_n$9 of $\mathrm{RFR}p$0, the pushforward argument shows the lifted class has strictly smaller $\mathrm{RFR}p$1-divisibility. This produces the mod-$\mathrm{RFR}p$2 version needed for the $\mathrm{RFR}p$3 application.

One should note the hypotheses: the unwrapping machinery relies on minimality of the immersed representative and on the planarity hypothesis in an essential way (the separation lemma and the spanning of $\mathrm{RFR}p$4 by arc characters both fail without it).

Boundary control

To promote the topological tower to a group-theoretic one compatible with the semidirect product structure, the authors prove a boundary lifting theorem. Suppose $\mathrm{RFR}p$5 acts isotopically on a planar surface $\mathrm{RFR}p$6 (by mapping classes fixing basepoint and boundaries pointwise). Such an action acts trivially on $\mathrm{RFR}p$7, hence preserves any arc character $\mathrm{RFR}p$8 and the corresponding cover subgroup $\mathrm{RFR}p$9. For each boundary component $p$0, the displacement exponent $p$1 measuring how canonical lifts translate marked points in the infinite cyclic cover is shown to be a well-defined homomorphism, independent of the chosen representative. Setting

$p$2

yields a finite-index normal subgroup whose elements fix every boundary component of the cover pointwise, inducing an isotopical action on $p$3. Moreover, the two-step sequence

$p$4

is a partial $p$5 tower (and a partial $p$6 tower when $p$7, since $p$8 is then elementary abelian). The key point is that each $p$9 vanishes on $A_n$0, so the radical containment required by the tower definition holds automatically.

Semidirect products and the main theorem

Combining these ingredients gives the stability result: if $A_n$1 is finitely generated $A_n$2 (resp. $A_n$3) acting isotopically on a compact planar surface $A_n$4, then $A_n$5 is $A_n$6 (resp. $A_n$7). Elements with nontrivial $A_n$8-component are detected by pulling back a detecting tower for $A_n$9; pure kernel elements $B_n = C_n$0 are detected by realizing the planar tower of covers and applying boundary control at each stage, so that at the terminal stage $B_n = C_n$1 fixes all boundary components, acts trivially on $B_n = C_n$2, and the split homology decomposition detects $B_n = C_n$3. The $B_n = C_n$4 case requires the mod-$B_n = C_n$5 tower theorem together with a final quotient by the kernel of reduction to $B_n = C_n$6.

Applying this inductively along the Fadell–Neuwirth splitting $B_n = C_n$7 — where $B_n = C_n$8 for a planar surface and $B_n = C_n$9 acts isotopically via the identification $RFRS$00 — establishes that $RFRS$01 is $RFRS$02, indeed $RFRS$03 for every prime $RFRS$04. The corollary for Artin groups follows from known virtual embeddings: type $RFRS$05 into braid groups of type $RFRS$06, type $RFRS$07 into type $RFRS$08 via the Crisp–Paris isomorphism, and type $RFRS$09 virtually into $RFRS$10 through its embedding in the punctured-sphere mapping class group, using that $RFRS$11 passes to subgroups.

A useful contrast drawn in the introduction: mapping class groups of surfaces are not virtually $RFRS$12 in general — fibered 3-manifold groups embed in punctured mapping class groups, and the mapping torus of a Dehn twist on a closed surface of genus at least 2 has fundamental group that is not virtually $RFRS$13. So the positive result is specific to the braid/pure-braid setting rather than a general phenomenon about mapping class groups.

Limitations and open questions

The paper resolves the virtual $RFRS$14 question for braid groups but leaves the broader questions open. Whether full braid groups, or Artin groups in general, are virtually special remains open (Haglund–Wise Problem 9.4, Wise Problem 13.4); since virtually special implies virtually $RFRS$15, the results here constitute necessary-condition evidence only, and no virtually special structure for $RFRS$16 is constructed. The list of Artin groups covered ($RFRS$17, $RFRS$18, $RFRS$19, $RFRS$20, $RFRS$21) is limited to those admitting virtual embeddings into (pure) braid groups or being virtually special already; other types, including most spherical and affine types beyond these, are untouched. The method also depends structurally on planarity of the fiber surfaces in the Fadell–Neuwirth splitting, and the paper does not address whether analogous arc-cyclic tower constructions exist for non-planar fibers. Finally, whether braid groups themselves could be virtually $RFRS$22 for some prime (rather than merely virtually $RFRS$23) is not settled by the argument, which passes through the finite-index subgroup $RFRS$24.

Conclusion

This paper answers Agol's modified question affirmatively: pure braid groups are $RFRS$25, and in fact $RFRS$26 for every prime $RFRS$27. The proof introduces a planar curve-unwrapping technique under arc-cyclic covers, together with a boundary control mechanism for isotopical actions, yielding a general stability theorem for semidirect products $RFRS$28 over planar surfaces. Beyond braid groups, the transfer to Artin groups of types $RFRS$29, $RFRS$30, $RFRS$31, $RFRS$32, and $RFRS$33 supplies concrete partial progress toward the virtual specialness conjecture for Artin groups, while leaving the general case open.