Partial Torelli Subgroups

• Partial Torelli subgroups are defined as subgroups that act trivially on selected portions of homological or nilpotent data, distinguishing them from full Torelli groups.
• They are constructed via kernel conditions from abelian or nonabelian markings and analyzed through Johnson and Magnus filtrations to ensure homological stability.
• These groups exhibit intricate presentations and infinite hierarchies, with finite presentability established in free group contexts and open questions remaining for mapping class groups.

A partial Torelli subgroup is a subgroup of a mapping class group, automorphism group of a free group, or related objects, that acts trivially on a specified subobject of the first homology or a nilpotent quotient of the fundamental group, but not necessarily on the entire homological or nilpotent structure. Such subgroups interpolate between the full group and the Torelli group (which acts trivially on all first homology), and are also constructed using kernels associated with markings or crossed homomorphisms. They have emerged as key objects in the study of homological stability, presentations, and representation theory of large discrete groups associated to surfaces, free groups, or 3-manifolds.

1. Definitions and Structure of Partial Torelli Subgroups

Partial Torelli subgroups arise by enforcing that only a specific part of the homological or nilpotent data is preserved under the action of the group. Given a surface $\Sigma_{g,b}$, with mapping class group $\mathrm{Mod}(\Sigma_{g,b})$, and a surjective homomorphism $\mu: H_1(\Sigma_{g,b}) \to A$ (where $A$ is a finitely generated abelian group), the abelian partial Torelli subgroup is

$$\mathcal{I}(\Sigma_{g,b}; \mu) = \ker\big( \mathrm{Mod}(\Sigma_{g,b}) \to \operatorname{Aut}(H_1(\Sigma_{g,b}) / \ker \mu) \big),$$

that is, those mapping classes acting trivially on the image under $\mu$ (Putman, 2019).

Nonabelian variants use a marking $\mu: \pi_1(\Sigma_{g,1}) \to \Lambda$ for a finite group $\Lambda$, and define

$$\mathcal{I}(\Sigma_{g,1}; \mu) = \ker\big( \mathrm{Mod}(\Sigma_{g,1}) \to \operatorname{Aut}(\pi_1(\Sigma_{g,1}) / \ker \mu) \big).$$

For $\operatorname{Aut}(F_n)$, partial Torelli subgroups arise by fixing some summands of the free abelianization: for example, the row-stabilizer for $d\le n$ is

$$K_{n,d} = \ker( \operatorname{Aut}(F_n) \to GL_n(\mathbb{Z}) / U_{n,d} )$$

where $U_{n,d}$ is the subgroup of $GL_n(\mathbb{Z})$ with the first $d$ rows those of the identity matrix. Analogously, column-stabilizers can be defined (Ershov, 4 Jan 2026).

In the context of filtrations, such as the Johnson or Magnus filtrations, higher terms define “deeper” partial Torelli subgroups: for instance, the kernel of the action on successive lower central or derived quotients of $\pi_1$ is a “partial Torelli” subgroup with respect to those nilpotent or solvable quotients (McNeill, 2013, Hadari, 2019).

2. Filtrations, Homomorphisms, and the Role of Johnson Theory

Partial Torelli subgroups appear naturally via filtrations:

• Johnson filtration: For $\Gamma = \mathrm{Mod}(\Sigma_g)$ or $\operatorname{Aut}(F_n)$, set $I = I_1$ (the standard Torelli), $L_1 = [\pi_1,\pi_1]$, and inductively $L_{k+1} = [\pi_1, L_k]$. Then $$I_{k+1} = \ker( \Gamma \to \operatorname{Aut}( \pi_1 / L_{k+1} ))$$ (Hadari, 2019).
• Magnus filtration: The derived series on $F = \pi_1(S_{g,1})$ leads to $$M_k(S) = \ker( \mathrm{Mod}(S) \to \operatorname{Aut}(F / F^{(k)}) )$$ with $F^{(k+1)} = [F^{(k)}, F^{(k)}]$ (McNeill, 2013).

Associated to these filtrations are Johnson-type homomorphisms that identify abelian quotients, e.g., the classical Johnson homomorphism

$$\tau: I_{g,p}^b \to \Lambda^3 H_1(\Sigma_{g,p}^b) \ \text{or} \ (\Lambda^3 H_1) / H_1$$

and analogous objects in other contexts (Holden, 3 Sep 2025). These yield graded pieces that can be analyzed using symplectic or $SL_g(\mathbb{Q})$-representation theory.

3. Homological Stability and Simplicial Complexes

One of the defining properties of partial Torelli subgroups is that they admit homological stability: for fixed “markings” $\mu$, the inclusions $\Sigma_{g,1} \to \Sigma_{g+1,1}$ induce isomorphisms on $H_k$ in large genus $g$ depending on the rank of the marking:

• For abelian markings $\mu: H_1 \to A$,

$$H_k(\mathcal{I}(\Sigma_{g,1}; \mu)) \to H_k(\mathcal{I}(\Sigma_{g+1,1}; \mu'))$$

is an isomorphism for $$g \geq (\mathrm{rank}\,A + 2)k + (2\,\mathrm{rank}\,A + 2)$$ (Putman, 2019).

• For nonabelian markings $\mu: \pi_1 \to \Lambda$, $\Lambda$ finite, similar results hold with $|\Lambda|$ replacing $\mathrm{rank}\,A$.

These results are proved by constructing highly connected simplicial (or semisimplicial) complexes—such as the complex of "vanishing" tethered subsurfaces, vanishing loop complexes, and their order-preserving double-tethered analogues—that enforce stabilization and vanishing conditions corresponding to the group action (Putman, 2019).

4. Presentations, Finiteness, and Combinatorial Aspects

Partial Torelli subgroups in automorphism groups of free groups exhibit favorable finiteness properties, in contrast to the longstanding open finite presentability problem for IA$_n$ (the classical Torelli for $\operatorname{Aut}(F_n)$, kernel of $GL_n(\mathbb{Z})$-action). Specifically, for the universal central extension $\tilde{\mathrm{Aut}}^+(F_n)$ and $n$ sufficiently larger than $d$, both the row- and column-stabilizer partial Torelli subgroups are finitely presented:

$$\text{For } n \ge d+115,\quad \tilde{K}_{n,d} = \rho_{ab}^{-1}(U_{n,d}) \text{ is finitely presented}$$

with the bound improved to $n\ge26$ when $d=1$ (Ershov, 4 Jan 2026).

The proof utilizes a “peak-reduction” van Kampen diagrammatic argument, with a Brown/Renz-type van Kampen criterion for subgroup finite presentability, a refined order on the coset space $G/H$, and exhaustive local analysis of moves in the Schreier graph. Notably, these groups are finitely generated by explicit Nielsen/Weyl generators, with only finitely many relations needed in each case (Ershov, 4 Jan 2026).

However, for mapping class group partial Torelli subgroups, finite presentability is generally unresolved, though stability results for homology are available.

5. Representations, Kernels, and (Non-)Detectability by Homology

Partial Torelli subgroups are intrinsically related to the failure of homological representations to separate deeper subgroups. Given any finite characteristic cover $\pi: \tilde{\Sigma} \to \Sigma$, the associated homological representation

$$\rho: \Gamma \to GL(H_1(\tilde{\Sigma};\mathbb{Z}))$$

collapses the distinction between the successive Johnson subgroups $I_k$ and the Torelli $I=I_1$: for $k\ge2$, the image $\rho(I_k)$ is a finite-index subgroup of $\rho(I)$, even though $I_k$ lies in $I$ with infinite index (Hadari, 2019). Therefore, no such representation can be faithful on $I_k$; all finite-cover homological representations collapse the infinite hierarchy of partial Torelli (i.e., higher Johnson) subgroups to subgroups commensurable with the Torelli image.

This phenomenon is explained via the $Sp$-module structure of the Johnson and Magnus quotients and nilpotence in the deeper steps.

6. Extensions, Cohomology, and Kernel Subgroups

Certain partial Torelli subgroups are kernels of crossed homomorphisms (Earle–Morita and Chillingworth subgroups) associated to Heisenberg group actions (Blanchet et al., 2023). For surfaces with boundary, these subgroups are realized as kernels:

• Earle–Morita subgroup: $$\operatorname{Mor}(\Sigma) = \ker \delta \subset \operatorname{Mod}(\Sigma)$$
• Chillingworth subgroup: $\operatorname{Chill}(\Sigma) = \ker e \cap T$

In the context of twisted Heisenberg homology representations, restriction to these subgroups yields genuine (untwisted) linear representations; the projective ambiguity, present on the full Torelli or mapping class group, vanishes on these subgroups. For the infinite-dimensional Schrödinger representation, the Earle–Morita subgroup admits honest linear representations without nontrivial central extension (Blanchet et al., 2023). These partial Torelli subgroups are also of infinite index and infinite rank for $g \ge 2$.

In the setting of handlebodies, analogues of partial Torelli subgroups are defined using actions on boundary vs. interior homology, and explicit Johnson-type homomorphisms and their SL$_g(\mathbb{Q})$-module structures are fully described, including cup-product maps in cohomology (Holden, 3 Sep 2025).

7. Higher-Order Magnus Subgroups and Infinite Generation

The higher-order Magnus filtration $M_k(S)$ on the mapping class group kernel of the Magnus representation provides an infinite filtration of the Torelli group by successively smaller partial Torelli subgroups, each defined as the kernel of the action on $F / F^{(k)}$ (derived series):

$$M_{k}(S) = \ker \big( \mathrm{Mod}(S) \to \operatorname{Aut}(F / F^{(k)}) \big)$$

with $M_1(S)$ the Torelli, $M_2(S)$ the Magnus kernel, and so on (McNeill, 2013). These have quotients $M_k(S)/M_{k+1}(S)$ that surject onto infinite-rank torsion-free abelian groups and contain free-group lower central series quotients. This filtration emphasizes the persistence and largeness of partial Torelli-type subgroups even far down the filtration ladder; in particular, none of these subgroups collapses to finite or trivial in low genus or for moderate $k$ (McNeill, 2013).

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References:

• Homological stability and core definitions: (Putman, 2019)
• Subgroup separability and Johnson filtration: (Hadari, 2019)
• Partial Torelli in $\operatorname{Aut}(F_n)$ and finite presentability: (Ershov, 4 Jan 2026)
• Heisenberg crossed homomorphism kernels: (Blanchet et al., 2023)
• Handlebody Torelli and Johnson/cup product structure: (Holden, 3 Sep 2025)
• Higher order Magnus kernels and infinite generation: (McNeill, 2013)