Hyperelliptic Torelli Group

• Hyperelliptic Torelli group is a subgroup of the mapping class group defined by commuting with a fixed hyperelliptic involution and acting trivially on surface homology.
• It is generated by Dehn twists about s-invariant separating curves and connected to braid groups via the Birman–Hilden correspondence.
• Its cohomological dimension is g–1 with infinitely generated homology, highlighting complex topological and algebraic structures.

The hyperelliptic Torelli group, denoted $\mathcal{SI}_g$ for a closed oriented surface $\Sigma_g$ of genus $g$, is the subgroup of the mapping class group $\mathrm{Mod}(\Sigma_g)$ consisting of mapping classes that commute with a fixed hyperelliptic involution $s$ and act trivially on the homology group $\mathrm{H}_1(\Sigma_g; \mathbb{Z})$. This group occupies a central position in low-dimensional topology and algebraic geometry, encoding symmetries of curves and their moduli spaces that preserve both the hyperelliptic structure and integral homology.

1. Definition and Basic Properties

A hyperelliptic involution $s$ on $\Sigma_g$ is an orientation-preserving homeomorphism of order two such that the quotient $\Sigma_g/s$ is a $2$-sphere with $2g+2$ branch points. The mapping class group $\mathrm{Mod}(\Sigma_g)$ consists of isotopy classes of orientation-preserving self-homeomorphisms. The Torelli group, $\mathcal{I}_g$, is the kernel of the natural symplectic representation $$\rho: \mathrm{Mod}(\Sigma_g) \to \mathrm{Sp}(2g, \mathbb{Z})$$ induced by action on the first homology. The centralizer $\mathrm{Comm}(s)$ of $s$ in $\mathrm{Mod}(\Sigma_g)$ is the hyperelliptic mapping class group.

The hyperelliptic Torelli group is given by

$$\mathcal{SI}_g = \{ f \in \mathrm{Mod}(\Sigma_g) \mid f \circ s = s \circ f,\, \rho(f) = \operatorname{Id} \} = \mathrm{Comm}(s) \cap \mathcal{I}_g.$$

It sits within the inclusion hierarchy

$$\mathcal{SI}_g \leq \mathcal{K}_g \leq \mathcal{I}_g \leq \mathrm{Mod}(\Sigma_g),$$

where $\mathcal{K}_g$ is the Johnson kernel generated by Dehn twists about all separating curves. $\mathcal{SI}_g$ is generated by Dehn twists about $s$-invariant separating curves (Spiridonov, 18 Jan 2026, Brendle et al., 2012, Brendle et al., 2012).

2. Generating Sets and Birman–Hilden Correspondence

A simple closed curve $\gamma \subset \Sigma_g$ is $s$-invariant separating if $s(\gamma) = \gamma$ (setwise) and $\gamma$ disconnects $\Sigma_g$. Dehn twists $T_\gamma$ about all such curves generate $\mathcal{SI}_g$ for every $g \geq 1$.

Via the Birman–Hilden theorem, the centralizer $\mathrm{Comm}(s)$ (or symmetric mapping class group) corresponds to the braid group acting on the punctured sphere (for the covering induced by $s$). The kernel of the Burau representation at $t = -1$ coincides with the hyperelliptic Torelli group, and Dehn twists around symmetric separating curves correspond to squares of standard braid group generators encircling odd numbers of punctures (Brendle et al., 2012, Brendle et al., 2011).

3. Cohomological Dimension and Homology

The cohomological dimension of $\mathcal{SI}_g$ is $g-1$ for $g > 0$, established via actions on contractible complexes of symmetric cycles. In particular, for $g \geq 2$, the top homology group $H_{g-1}(\mathcal{SI}_g; \mathbb{Z})$ is infinitely generated (Brendle et al., 2011, Spiridonov, 18 Jan 2026). For genus $g = 3$, $H_2(\mathcal{SI}_3; \mathbb{Z})$ is infinitely generated and can be explicitly described in terms of "simple abelian cycles":

• Each cycle arises from a pair of disjoint $s$-invariant separating curves $(\gamma, \delta)$ whose Dehn twists commute, yielding an abelian cycle $A(T_\gamma,T_\delta)$ in $H_2(\mathcal{SI}_3; \mathbb{Z})$.
• These cycles correspond to orthogonal splittings $V = V_1 \oplus V_2 \oplus V_3$ of $H_1(\Sigma_3; \mathbb{Z})$ (where $V_i$ correspond to subsurfaces after separating), subject to an Arf invariant condition: $Arf(\omega_0|_{V_1}) = Arf(\omega_0|_{V_3}) = 1$ while $Arf(\omega_0|_{V_2}) = 0$ for a specified quadratic form $\omega_0$.
• These simple abelian cycles are linearly independent in $H_2(\mathcal{SI}_3; \mathbb{Z})$ (Spiridonov, 18 Jan 2026).

4. Exact Sequences, Birman Sequences, and Algorithmic Factorization

Exact sequences play a fundamental role in understanding the structure and presentations of $\mathcal{SI}_g$. For the punctured surface setting, the Birman exact sequence splits for the hyperelliptic Torelli group: for S_g with a marked point, the forgetful homomorphism is an isomorphism (Brendle et al., 2011). For pairs of points exchanged by the involution, a split extension is realized whose kernel is an infinite-rank free group.

Algorithmic approaches have been developed for factoring arbitrary elements of $\mathcal{SI}_g$ into products of Dehn twists about symmetric separating curves. For the punctured disk model under the Birman–Hilden correspondence, factoring reduces to manipulations in the free group setting, with an explicit algorithm for decomposing elements into twists about genus $1$ and $2$ curves (Brendle et al., 2012). Notably, there exist worked examples demonstrating factorization procedures for commutators and higher-genus twists in terms of genus $1$ or $2$ separating twists.

5. Topological and Geometric Realizations

$\mathcal{SI}_g$ is fundamental to the study of the topology of moduli spaces, especially the Torelli space and its hyperelliptic locus. The Torelli space $T_g$ is the moduli space of genus-$g$ curves with symplectic homology bases. The hyperelliptic locus $\mathrm{Hyp}_g \subset T_g$ consists of points corresponding to hyperelliptic curves. Each component of $\mathrm{Hyp}_g$ is K$(\mathcal{SI}_g,1)$; the fundamental group is $\mathcal{SI}_g$ (Kordek, 2015). For $g \geq 3$, the rational homology $H_2(\mathcal{SI}_g; \mathbb{Q})$ and the topology of the locus are infinitely generated and have infinite codimension, indicating that these spaces do not have the homotopy type of a finite CW complex.

In genus $3$, components of the hyperelliptic locus correspond to analytic divisors defined by vanishing thetanull functions in Siegel space $\mathfrak{h}_3$. Each such component is simply-connected and the third homology group is free abelian; higher homology vanishes. The possibility that the second homology is also free abelian remains open, which would imply a wedge-of-spheres decomposition by Whitehead’s theorem (Kordek, 2015).

6. Analogues in Free Group Settings and Further Generalizations

The concept of hyperelliptic Torelli extends to free group automorphism groups via the period mapping on Culler–Vogtmann outer space $CV_n$: the "free group" hyperelliptic Torelli group $\mathrm{ST}(n)$ is the intersection of the centralizer of the canonical involution $x_i \mapsto x_i^{-1}$ with the Torelli subgroup of $\mathrm{Out}(F_n)$. $\mathrm{ST}(n)$ is generated by doubled-commutator transvections and underpins the topology of the hyperelliptic locus in Torelli space for graphs. The components become simply-connected when certain degenerate graphs are added (Bregman et al., 2016).

For higher genus $g \geq 3$, the group $\mathcal{SI}_g$ retains cohomological dimension $g-1$ but an explicit basis for $H_{g-1}(\mathcal{SI}_g; \mathbb{Z})$ remains unknown. It is conjectured that simple abelian cycles associated to $g-1$ pairwise disjoint $s$-invariant separating curves generate an infinite-rank free summand, but minimal dependencies may arise from intersection patterns and higher-rank Arf invariants (Spiridonov, 18 Jan 2026).

7. Connections to Burau Representation and Related Structures

The kernel of the Burau representation of the braid group $B_n$ at $t = -1$ is naturally isomorphic to the hyperelliptic Torelli group for appropriately chosen surfaces and is central to the Birman–Hilden dictionary. Finiteness properties, such as infinite generation and non-finite presentability of $\mathcal{SI}_g$ for $g \geq 3$, extend to the corresponding kernels of the Burau representation, whose cohomological dimension is $\lfloor n/2 \rfloor$ (Brendle et al., 2011, Brendle et al., 2011).

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These interconnected perspectives reveal the hyperelliptic Torelli group as a rich and multi-faceted object, central to various realms in geometric group theory, low-dimensional topology, and the study of moduli spaces. Its algebraic structure, generating sets, homological properties, and topological avatars underscore its significance and open further avenues for exploration in both classical and free settings.