bibkey: banerjee2025twiststabilizer
authors: Ishan Banerjee
year: 2025
title: A pi_1 obstruction to having finite index monodromy and an unusual subgroup of infinite index in Mod(Sigma_g)
doi: 10.1007/s00208-025-03213-7
url: https://doi.org/10.1007/s00208-025-03213-7
claim: Definition 1 defines the mapping-class action on target-conjugacy classes of surface-group epimorphisms to arbitrary groups; Proposition 3 asserts that a nonseparating Dehn twist stabilizes such a class exactly when its curve lies in the epimorphism kernel.
strata_touched: []
license: citation-only
triage: anchor

Nonseparating twist stabilizers

Verified locator

Ishan Banerjee, Mathematische Annalen 392 (2025), 5045-5064. Published July 4, 2025. DOI: 10.1007/s00208-025-03213-7. https://doi.org/10.1007/s00208-025-03213-7

Source scope

Definition 1 allows an arbitrary target group G and a surjection phi from a closed surface group. Mapping classes act by precomposition on epimorphisms modulo conjugation by G. Proposition 3 asserts that a nonseparating Dehn twist preserves the conjugacy class of phi if and only if its curve lies in ker(phi). It imposes no abelianness, centralizer condition or geometric realization of G.

Correspondence to the theory

Sections 22-23 of SURFACE_CONGRUENCE_OBSERVERS give a genus-two counterexample and the corrected centralizer criterion. Use the geometric presentation [a,b][c,d]=1 and the nonseparating twist fixing a,c,d and sending b to ba. The identity [a,ba]=[a,b] preserves the surface relator.

Let D8 have r^4=s^2=1 and srs=r^-1. The assignment (a,b,c,d) -> (r^2,s,r,1) preserves the relator and is onto. Precomposition with the twist is conjugation by r on the entire representation, but phi(a)=r^2!=1. Thus stabilization up to conjugacy does not imply membership in ker(phi). The implication from kernel membership to stabilization remains valid. The inverse twist gives the same counterexample. Changing the based lift by an inner automorphism preserves the target-conjugacy-class condition.

With A=phi(a), B=phi(b), and S generated by the images of generators fixed by T, stabilization under T^n is equivalent to the existence of z in C_G(S) with zBz^-1=BA^n. Exact marked invariance is A^n=1. The stated equivalence is restored when C_G(S) is contained in the center of G. A torsion-free analogue uses the integral Heisenberg group [X,Y]=Z and images (Z,Y,X,1).

Boundaries

The counterexample concerns the arbitrary-group assertion of Proposition 3. It does not establish failure of the article’s geometric monodromy theorems: the target epimorphism is not asserted to arise from an ample-curve inclusion in a smooth projective surface. Proposition 4’s use of Proposition 3 requires a justification respecting that geometric scope.

One stabilized representation differs from the action on all representations in a finite characteristic quotient. The D8 representation (r,1,s,1) detects the same twist. Neither this counterexample nor the centralizer criterion resolves the general curve-orbit congruence conjecture.