Isometric embeddings of surfaces for scl

TL;DR

This paper proves that geometric injective morphisms of free groups preserve stable commutator length and extends this isometry property to subsurfaces within surfaces, using admissible surface techniques.

Contribution

It establishes isometric embeddings for stable commutator length under geometric injections and generalizes to relative Gromov seminorms for subsurfaces.

Findings

Geometric injective morphisms are isometric for stable commutator length.

Subsurface chains embed isometrically into larger surface homology groups.

Admissible surface analysis underpins the isometry proofs.

Abstract

Let $\varphi:F_1\to F_2$ be an injective morphism of free groups. If $\varphi$ is geometric (i.e. induced by an inclusion of oriented compact connected surfaces with nonempty boundary), then we show that $\varphi$ is an isometric embedding for stable commutator length. More generally, we show that if $T$ is a subsurface of an oriented compact (possibly closed) connected surface $S$, and $c$ is an integral $1$-chain on $\pi_1T$, then there is an isometric embedding $H_2(T,c)\to H_2(S,c)$ for the relative Gromov seminorm. Those statements are proved by finding an appropriate standard form for admissible surfaces and showing that, under the right homology vanishing conditions, such an admissible surface in $S$ for a chain in $T$ is in fact an admissible surface in $T$.