For which $n$ and $q$ is the $A_n(q)$ complex reflection group a lattice in $U(r,s)$? (McMullen, Question 5.6)

McMullen (2013) asked: for which values of $n$ and $q$ is the $A_n(q)$ complex reflection group - the image of the braid group $B_n$ acting by monodromy on the $q$-eigenspace of the first cohomology of the cyclic branched cover $y^d = \prod_{i=1}^n (x - b_i)$ of the Riemann sphere, equivalently the reduced Burau representation specialized at the root of unity $q$ - a lattice in the indefinite unitary group $U(r,s)$? The question is equivalent to arithmeticity (finite index in an explicit arithmetic group $U(\Lambda_{n,q})$) in signatures with $r, s \ge 2$, and to a lifting question for $B_n$ into the centralizer of the deck transformation in the integral symplectic group. Venkataramana (Annals of Mathematics, 2014) resolved the degrees $d \in \{3,4,6\}$ for all $n$ and showed the image is never a lattice for $d \notin \{2,3,4,6\}$ with $n \ge 2d+1$ branch points; Menet-Nguyen (2023) extended arithmeticity of the pure braid monodromy down to $n \ge d+1$ under mild hypotheses. The remaining open core comprises the sporadic discrete pairs with $d \in \{8,10,12\}$ and signature $r \ge 2$ - fifteen $(n, k/d)$ pairs with $n \le d$ and their complex conjugates, where the ambient group $U(\Lambda_{n,q})$ is an arithmetic lattice but finite-index containment of the braid image is unknown - together with the pairs $d \notin \{2,3,4,6\}$, $n \le 2d$ outside the discreteness table, where even discreteness of the image is undetermined, and the associated thinness question (whether images can be of infinite index in the integral points of their Zariski closure).

Background

Let $B_n$ denote the braid group on $n$ strands: the fundamental group of the configuration space of $n$ distinct points in the complex plane, with generators $\tau_1, \dots, \tau_{n-1}$ subject to the braid relations. Fix $d \ge 2$ and $n \ge 3$, choose distinct points $b_1, \dots, b_n \in \mathbb{C}$, and let $X$ be the cyclic branched cover of the Riemann sphere $\mathbb{P}^1$ defined by $y^d = \prod_{i=1}^n (x - b_i)$. The curve $X$ carries a deck transformation $T(x, y) = (x, \zeta_d y)$ of order $d$, where $\zeta_d$ is a primitive $d$-th root of unity, and moving the branch points along loops induces a monodromy representation of $B_n$ on the first cohomology $H^1(X)$. Because this action commutes with $T$, the space $H^1(X)$ splits into eigenspaces $H^1(X)_q$ indexed by $d$-th roots of unity $q$, and the restriction of the intersection pairing to each eigenspace is a Hermitian form of some signature $(r, s)$; the projected representation takes values in the unitary group $\rho_q : B_n \to U(H^1(X)_q) \cong U(r,s),$ where $U(r,s)$ is the group of complex linear automorphisms preserving a Hermitian form of signature $(r,s)$. Up to conjugation, $\rho_q$ is the classical reduced Burau representation of the braid group - the reduction of the Burau representation, a one-parameter family of representations studied since the 1930s via the Magnus expansion of a free group - evaluated at the parameter $t = q$. Its image is the complex reflection group denoted $A_{n-1}(q)$ (in the indexing used in McMullen's introduction; the same family is written $A_n(q)$ when the braid group has $n+1$ strands, the convention of Question 5.6 itself). Here a complex reflection is a finite-order linear map that fixes a hyperplane pointwise, and $A_{n-1}(q)$ is the image of the Artin group of type $A_{n-1}$ in the natural representation of the Hecke algebra at parameter $q$ (the Hecke algebra being the quantum deformation of the group algebra of the symmetric group). A lattice in a real Lie group $G$ is a discrete subgroup $\Gamma \subset G$ such that the quotient $G/\Gamma$ has finite volume for the Haar measure. A subgroup is arithmetic, or an arithmetic group, when it is commensurable with the group of integral points of some $\mathbb{Q}$-algebraic group, where commensurable means the two groups share a common subgroup of finite index in both. By the Margulis arithmeticity theorem, every lattice in $U(r,s)$ with $r, s \ge 2$ is arithmetic, whereas in rank one ($r = 1$) non-arithmetic lattices exist; the primary source settles the rank-one cases using the classical Deligne-Mostow theory of hypergeometric monodromy as presented there, which realizes monodromy groups of certain hypergeometric differential equations as lattices in $U(1,s)$ via an equivalent orbifold criterion. McMullen (2013) observed that $\rho_q(B_n)$ preserves the natural module $\Lambda_{n,q} = H^1(X, \mathbb{Z}[q])_q$ over the ring $\mathbb{Z}[q]$ obtained by adjoining $q$ to the integers, so $\rho_q(B_n) \subset U(\Lambda_{n,q}) \subset U(r,s)$, and classified exactly when the ambient group $U(\Lambda_{n,q})$ is discrete: by Theorem 7.1 and Table 9 of that paper, for $n > 12$ only when $\mathbb{Z}[q]$ is a discrete subring of $\mathbb{C}$, that is $d \in \{2,3,4,6\}$, together with an explicit finite list of sporadic pairs $(n, k/d)$ (writing $q = \zeta_d^{-k}$), including $(6,3/8)$ through $(8,3/8)$, $(7,3/10)$ through $(10,3/10)$, and $(5,5/12)$ through $(12,5/12)$, plus complex conjugates. In signature $(1,s)$, McMullen proved that $\rho_q(B_n)$ is a lattice whenever it is discrete (Theorem 10.3 and Corollary 10.4, via the Deligne-Mostow framework, in the orbifold form presented in that paper), giving 24 arithmetic cases, and the $n = 3$ rows are arithmetic triangle groups, settled by the classical classification of arithmetic hyperbolic triangle groups. The paper also poses a companion topological question (Question 11.1): when is the natural map $B_n \to Sp(X)_d^T$ almost onto, where $Sp(X)_d^T$ denotes the subgroup of the integral symplectic group of $X$ commuting with the deck transformation and restricted to the $d$-primary part, and almost onto means the image has finite index; Proposition 11.3 proves this equivalent to $\rho_q(B_n)$ having finite index in $U(\Lambda_{n,q})$ for some primitive $d$-th root of unity $q$. Venkataramana (Annals of Mathematics, 2014) proved that for $d \in \{3,4,6\}$ and every $n$ the Burau image at a primitive $d$-th root is an arithmetic subgroup of the integral unitary group, and that for $d \notin \{2,3,4,6\}$ with at least $2d+1$ branch points the image is a lattice in the product of the unitary groups over all archimedean completions while its projection to any single factor is never a lattice (it is dense in the special unitary factor). Venkataramana (Inventiones mathematicae, 2014) established the analogous arithmeticity for pure braid monodromy with exponents coprime to $d$ in the same large-$n$ regime. Menet-Nguyen (2023) determined the Zariski closure of the pure braid monodromy and extended its arithmeticity to essentially $n \ge d+1$ branch points under explicit mild hypotheses, while posing the complementary thinness question - whether the images can be thin, meaning of infinite index in the integral points of their Zariski closure - as open. The lattice question thus survives precisely in degrees $d \notin \{2,3,4,6\}$ with at most $2d$ branch points, most sharply in the sporadic discrete pairs with $d \in \{8,10,12\}$ and signature $r \ge 2$.

Problem Statement

The problem is Question 5.6 of McMullen, "Braid groups and Hodge theory" (Mathematische Annalen 355 (2013), 893-946), stated verbatim: $\text{For what values of } n \text{ and } q \text{ is the } A_n(q) \text{ reflection group a lattice in } U(r,s)?$ Here $n \ge 3$ is the number of branch points, and all pairs $(n, q)$ reported below are in this convention (so the symbol $A_n(q)$ denotes the image of $\rho_q(B_n)$, i.e. the primary source's $A_{n-1}(q)$; the symbol $A_n(q)$ in the title and in the verbatim Question 5.6 above follows instead the source's convention of attaching $A_n(q)$ to a braid group with $n+1$ strands, so it denotes what this task writes $A_{n+1}(q)$) of the cyclic cover $y^d = \prod_{i=1}^n (x - b_i)$ of the Riemann sphere, and $q$ ranges over roots of unity with $q \ne 1$ (write $q = \zeta_d^{-k}$ with $k$ coprime to $d$); $A_n(q)$ is the image of the braid monodromy representation $\rho_q$ on the $q$-eigenspace of the first cohomology - equivalently, in the primary source's indexing, the complex reflection group $A_{n-1}(q)$ obtained from the reduced Burau representation specialized at $t = q$ - and $(r,s)$ is the signature of the Hermitian form it preserves. The question admits two equivalent forms proved in the same paper: for which $(n, q)$ does $\rho_q(B_n)$ have finite index in the arithmetic group $U(\Lambda_{n,q})$, where $\Lambda_{n,q} = H^1(X, \mathbb{Z}[q])_q$; and for which $(n, d)$ is the natural map $B_n \to Sp(X)_d^T$ almost onto, i.e. of finite index (Question 11.1, equivalent by Proposition 11.3). By the partial results now available, the unresolved regime is $d \notin \{2,3,4,6\}$ with at most $2d$ branch points; the sharpest open instances are the fifteen sporadic discrete pairs with $d \in \{8,10,12\}$ and signature $r \ge 2$, namely $(n, k/d) = (6,3/8), (7,3/8), (8,3/8), (7,3/10), (8,3/10), (9,3/10), (10,3/10), (5,5/12), (6,5/12), (7,5/12), (8,5/12), (9,5/12), (10,5/12), (11,5/12), (12,5/12)$, together with their complex conjugates, where $U(\Lambda_{n,q})$ is an arithmetic lattice and the question is whether the braid image has finite index in it or is instead thin (of infinite index in $U(\Lambda_{n,q})$), and the pairs outside the discreteness table in the same range, where even discreteness of the image is undetermined. A complete answer must decide, for every pair $(n, q)$, whether $A_n(q)$ is a lattice in $U(r,s)$ and, when it is, whether it is arithmetic; this also settles the equivalent lifting question (Question 11.1). Whether an image that is not a lattice is thin or dense is the separate thinness question of Menet-Nguyen, which is not decided by lattice status alone in the non-discrete regime; it lies within the open core described above but is separate from the deliverable, so a complete answer to the lattice question need not decide whether a non-lattice image is thin or dense.

References

1. Curtis T. McMullen, "Braid groups and Hodge theory", Mathematische Annalen 355 (2013), no. 3, 893-946. DOI: 10.1007/s00208-012-0804-2, https://doi.org/10.1007/s00208-012-0804-2
2. T. N. Venkataramana, "Image of the Burau representation at d-th roots of unity", Annals of Mathematics (2) 179 (2014), no. 3, 1041-1083. DOI: 10.4007/annals.2014.179.3.4, arXiv:1205.6543, https://arxiv.org/abs/1205.6543
3. T. N. Venkataramana, "Monodromy of cyclic coverings of the projective line", Inventiones mathematicae 197 (2014), no. 1, 1-45. DOI: 10.1007/s00222-013-0477-9, arXiv:1204.4778, https://arxiv.org/abs/1204.4778
4. Gabrielle Menet and Duc-Manh Nguyen, "Representations of braid groups via cyclic covers of the sphere: Zariski closure and arithmeticity", preprint (2023; v3 revised November 2024), arXiv:2310.10401, https://arxiv.org/abs/2310.10401