Faithfulness of the Bardakov-Bellingeri conjugation representations of surface braid groups on free groups

Artin's classical theorem embeds the n-strand braid group $B_n$ faithfully into the automorphism group $\operatorname{Aut}(F_n)$ of a free group, via the conjugation action of braids on the fundamental group of the punctured disk. Bardakov and Bellingeri (2007, published 2011) extended this construction to surface braid groups: inside a suitable mixed braid group there is a naturally defined free subgroup on which the surface braid group acts by conjugation, yielding representations into automorphism or outer automorphism groups of free groups. They proved faithfulness for oriented surfaces with boundary, both into $\operatorname{Aut}(U_{n-1,g,p})$ and, after projection, into $\operatorname{Out}(U_{n-1,g,p})$ (the latter for $p > 0$ when $g > 0$ and for $p > 2$ when $g = 0$). Two cases were left explicitly open: (i) the representation $\rho_W: B_n(N_{g,p}) \to \operatorname{Aut}(W_{n,g,p})$ for non-orientable surfaces $N_{g,p}$ of genus $g \ge 1$ with $p \ge 1$ boundary components and $n > 1$ strands; and (ii) the representation $\tilde\rho_V: B_{n-1}(\Sigma_g) \to \operatorname{Out}(V_{n-1,g})$ for closed orientable surfaces $\Sigma_g$ of genus $g \ge 1$ with $n > 1$. This problem asks to determine, in each of the two remaining surface classes, whether the Bardakov-Bellingeri representation is faithful, and to describe the kernel when it is not.

Background

Let $F_n$ denote the free group of rank $n$ and $\operatorname{Aut}(F_n)$, $\operatorname{Out}(F_n)$ its automorphism group and its outer automorphism group, the quotient of $\operatorname{Aut}(F_n)$ by the subgroup of inner automorphisms (conjugations). Artin's theorem represents $B_n$ as the subgroup of $\operatorname{Aut}(F_n)$ sending each generator $x_i$ to a conjugate of some $x_{s(i)}$ and fixing the product $x_1x_2\cdots x_n$; geometrically $B_n$ is the mapping class group of the disk with $n$ marked points (the group of isotopy classes of orientation-preserving homeomorphisms preserving the marked set) and acts on $\pi_1$ of the punctured disk, which is free. This problem concerns the extension of this picture to braid groups of surfaces.

For a surface $\Sigma$, the $n$-strand surface braid group $B_n(\Sigma)$ is the fundamental group of the configuration space of $n$ unordered points on $\Sigma$. Finite presentations are known: for a closed orientable surface $\Sigma_g$ of genus $g \ge 1$, $B_n(\Sigma_g)$ is generated by $\sigma_1,\dots,\sigma_{n-1}$ and $x_1,\dots,x_{2g}$ (the latter take a point around the handles), subject to braid relations, mixed relations, and the surface relation $[x_1^{-1},x_2][x_3^{-1},x_4]\cdots[x_{2g-1}^{-1},x_{2g}] = \sigma_1\sigma_2\cdots\sigma_{n-1}^2\cdots\sigma_2\sigma_1$; for a compact non-orientable surface $N_{g,p}$ of genus $g \ge 1$ with $p \ge 1$ boundary components, $B_n(N_{g,p})$ is generated by $\sigma_1,\dots,\sigma_{n-1}$, the crosscap loops $a_1,\dots,a_g$ and boundary loops $z_1,\dots,z_{p-1}$ of the fundamental group of $N_{g,p}$, with analogous relations (Bellingeri 2004).

The Bardakov-Bellingeri construction uses the mixed braid group $D_n(\Sigma) = \pi^{-1}(S_{n-1})$, the subgroup of $B_n(\Sigma)$ of braids whose underlying permutation fixes the last point (the last strand returns to its starting position), where $\pi: B_n(\Sigma) \to S_n$ records the underlying permutation. A Reidemeister-Schreier computation (the standard algorithm that produces generators for a subgroup of a presented group from coset representatives) on $D_n(\Sigma)$ produces distinguished generators: elements $\tau_1,\dots,\tau_{n-1}$ (conjugated forms of the squares $\sigma_k^2$, geometrically loops of a moving point around the fixed point), together with conjugated forms $w_r$ of the handle (or crosscap) loops and $\xi_j$ of the boundary loops. The subgroups they generate are free and carry the representation: $U_{n-1,g,p} = \langle\tau_1,\dots,\tau_{n-1}, w_1,\dots,w_{2g}, \xi_1,\dots,\xi_{p-1}\rangle$ of rank $n+2g+p-2$ in the orientable-with-boundary case, $W_{n,g,p} = \langle\tau_1,\dots,\tau_{n-1}, w_1,\dots,w_g, \xi_1,\dots,\xi_{p-1}\rangle$ of rank $n+g+p-2$ in the non-orientable case (a subgroup of $D_{n+1}(N_{g,p})$), and $V_{n-1,g} = \langle\tau_2,\dots,\tau_{n-1}, w_1,\dots,w_{2g}\rangle$ of rank $2g+n-2$, normal in $D_n(\Sigma_g)$, in the closed orientable case. The relevant surface braid group acts on each free subgroup by conjugation, with explicit generator-wise formulas.

For oriented surfaces with boundary the source proves: (Theorem 2) $\rho_U: B_{n-1}(\Sigma_{g,p}) \to \operatorname{Aut}(U_{n-1,g,p})$ is faithful for $g \ge 0$, $p > 0$, $n \ge 3$, by realizing the action as conjugation inside the mapping class group of the punctured surface and applying curve-intersection arguments; and (Theorem 3) the projected representation into $\operatorname{Out}(U_{n-1,g,p})$ is faithful for $p > 0$ when $g > 0$ and for $p > 2$ when $g = 0$, using that elements acting by inner automorphisms are central and that the relevant braid groups have trivial center. As corollaries, $B_m(\Sigma_{g,p})$ with $p > 0$ is residually finite (its nontrivial elements survive in some finite quotient), and Fox calculus (the free-differential calculus on group rings) yields a Burau-type linear representation from $\rho_U$. Two constructions could not be settled. (i) For $N_{g,p}$ non-orientable with boundary, $B_n(N_{g,p})$ acts on $W_{n,g,p}$, giving $\rho_W: B_n(N_{g,p}) \to \operatorname{Aut}(W_{n,g,p})$ with the formulas of the source's Theorem 4; every element of the image fixes the distinguished element $A = \tau_{n-1}^{-1}\cdots\tau_2^{-1}\tau_1^{-1}\xi_1\xi_2\cdots\xi_{p-1}w_1^2w_2^2\cdots w_g^2$ (the source's Proposition 5). (ii) For closed $\Sigma_g$, the subgroup $\bar B_{n-1}(\Sigma_g) = \langle\sigma_1,\dots,\sigma_{n-2}, x_1,\dots,x_{2g}\rangle$ of $D_n(\Sigma_g)$ acts by conjugation on $V_{n-1,g}$ by automorphisms (Proposition 7), but $\bar B_{n-1}(\Sigma_g)$ is not $B_{n-1}(\Sigma_g)$: the surface relation forces $\tau_1 = [w_1^{-1},w_2][w_3^{-1},w_4]\cdots[w_{2g-1}^{-1},w_{2g}]\tau_{n-1}^{-1}\cdots\tau_2^{-1}$ to lie in $V_{n-1,g}$, and $B_{n-1}(\Sigma_g) \simeq \bar B_{n-1}(\Sigma_g)/\langle\langle\tau_1\rangle\rangle$. The action therefore descends only to the outer level, giving $\tilde\rho_V: B_{n-1}(\Sigma_g) \to \operatorname{Out}(V_{n-1,g})$ (Proposition 8), whose formulas for the action of the generators $\sigma_1,\dots,\sigma_{n-2}, x_1,\dots,x_{2g}$ on the free generators are well-defined up to conjugacy. For comparison, already over the disk the projection $B_n \to \operatorname{Out}(F_n)$ has kernel the infinite cyclic center of $B_n$.

Geometrically the free subgroups are point-pushing-type subgroups (their generators are bounding-pair braids — multitwists along two curves enclosing a single puncture, also called spin maps — in the mapping class group of the punctured surface), so the open questions ask whether the conjugation action of the remaining surface braid groups on such subgroups is faithful - a surface-braid analogue of Artin's theorem. Faithfulness would embed $B_n(N_{g,p})$ into $\operatorname{Aut}(F_{n+g+p-2})$ and $B_{n-1}(\Sigma_g)$ into $\operatorname{Out}(F_{2g+n-2})$, paralleling the classical characterization of mapping class groups as outer automorphism groups of surface groups.

Problem Statement

Determine whether the two Bardakov-Bellingeri conjugation representations of surface braid groups, left open by the source, are faithful, and describe their kernels.

Component (1) - non-orientable surfaces with boundary. Let $N_{g,p}$ be a compact connected non-orientable surface of genus $g \ge 1$ with $p \ge 1$ boundary components and let $n > 1$. Inside the mixed braid group $D_{n+1}(N_{g,p})$ consider the free group $W_{n,g,p} = \langle\tau_1,\dots,\tau_{n-1}, w_1,\dots,w_g, \xi_1,\dots,\xi_{p-1}\rangle$ of rank $n+g+p-2$ generated by the Reidemeister-Schreier-type elements ($\tau_k$ conjugated forms of $\sigma_k^2$; $w_r$, $\xi_j$ conjugated forms of the crosscap and boundary generators $a_r$, $z_j$), as defined in Section 4 of the source [1, 2]. The group $B_n(N_{g,p})$ acts on $W_{n,g,p}$ by conjugation, giving the representation $\rho_W: B_n(N_{g,p}) \to \operatorname{Aut}(W_{n,g,p}) \cong \operatorname{Aut}(F_{n+g+p-2})$ defined on the generators $\sigma_i$ ($1 \le i \le n-1$), $a_r$ ($1 \le r \le g$), $z_j$ ($1 \le j \le p-1$) by the explicit formulas of the source's Theorem 4. Question: is $\rho_W$ injective? A complete answer determines the set of triples $(g,n,p)$ with $g \ge 1$, $n > 1$, $p \ge 1$ for which $\rho_W$ is faithful and describes $\ker\rho_W$ otherwise; note that every element of the image fixes the element $A = \tau_{n-1}^{-1}\cdots\tau_2^{-1}\tau_1^{-1}\xi_1\xi_2\cdots\xi_{p-1}w_1^2w_2^2\cdots w_g^2$, which constrains the possible image.

Component (2) - closed orientable surfaces. Let $\Sigma_g$ be a closed orientable surface of genus $g \ge 1$ and let $n > 1$. In $D_n(\Sigma_g)$ consider the free group $V_{n-1,g} = \langle\tau_2,\dots,\tau_{n-1}, w_1,\dots,w_{2g}\rangle$ of rank $2g+n-2$, normal in $D_n(\Sigma_g)$ and containing the element $\tau_1 = [w_1^{-1},w_2][w_3^{-1},w_4]\cdots[w_{2g-1}^{-1},w_{2g}]\tau_{n-1}^{-1}\cdots\tau_2^{-1}$. With $\bar B_{n-1}(\Sigma_g) = \langle\sigma_1,\dots,\sigma_{n-2}, x_1,\dots,x_{2g}\rangle$ acting on $V_{n-1,g}$ by conjugation (the source's Proposition 7) and $B_{n-1}(\Sigma_g) \simeq \bar B_{n-1}(\Sigma_g)/\langle\langle\tau_1\rangle\rangle$, the action descends to the representation $\tilde\rho_V: B_{n-1}(\Sigma_g) \to \operatorname{Out}(V_{n-1,g}) \cong \operatorname{Out}(F_{2g+n-2})$ given, up to conjugacy, on the generators of $B_{n-1}(\Sigma_g)$ by the formulas (S1)-(S11) of the source's Proposition 8. Question: is $\tilde\rho_V$ injective? A complete answer determines the pairs $(g,n)$ with $g \ge 1$, $n > 1$ for which it is faithful and describes $\ker\tilde\rho_V$ otherwise; in particular it identifies which nontrivial surface braids act as inner automorphisms of $V_{n-1,g}$.

The two components share one objective: completing the faithfulness picture begun by Artin for the disk and by the source for oriented surfaces with boundary, for the Bardakov-Bellingeri representations of surface braid groups as (outer) automorphisms of free groups, in the source's natural generality. Answers may treat the components separately; partial resolutions (specific genera or strand numbers) count insofar as they contribute to the complete regime determination.

References

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