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Isometric embeddings of surfaces for scl

Alexis Marchand

TL;DR

The paper develops a framework to compare stable commutator length and the relative Gromov seminorm under embeddings of surfaces, proving that for a $\pi_1$-injective subsurface $T\subset S$ with appropriate hypotheses, admissible surfaces in $S$ can be homotoped into $T$, yielding isometric embeddings of $H_2(T,c)$ into $H_2(S,c)$ and, consequently, isometric embeddings for $\mathrm{scl}$ and the relative Gromov norm. The core methodology introduces a standard (and perfect standard) form for admissible surfaces, combining incompressibility, transversality, connected links, and a non-folding/orientation-perfect framework to control how surfaces sit inside the ambient space. This approach yields precise isometry results (including in the closed-surface setting) and clarifies the interplay between extremal surfaces and quasimorphisms through Bavard duality, notably showing that rotation quasimorphisms respect restriction to subsurfaces under suitable injectivity hypotheses. By separating computation into the relative Gromov seminorm and a subsequent infimum step, the paper suggests a two-step strategy for scl calculations in more complex groups and connects these ideas to broader questions about the structure of scl norms and their dual objects. Overall, the results advance understanding of scl and relative seminorms in surface groups, with potential implications for extremal objects and dynamics on surface groups.

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 $π_1T$, 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 $T$.

Isometric embeddings of surfaces for scl

TL;DR

The paper develops a framework to compare stable commutator length and the relative Gromov seminorm under embeddings of surfaces, proving that for a -injective subsurface with appropriate hypotheses, admissible surfaces in can be homotoped into , yielding isometric embeddings of into and, consequently, isometric embeddings for and the relative Gromov norm. The core methodology introduces a standard (and perfect standard) form for admissible surfaces, combining incompressibility, transversality, connected links, and a non-folding/orientation-perfect framework to control how surfaces sit inside the ambient space. This approach yields precise isometry results (including in the closed-surface setting) and clarifies the interplay between extremal surfaces and quasimorphisms through Bavard duality, notably showing that rotation quasimorphisms respect restriction to subsurfaces under suitable injectivity hypotheses. By separating computation into the relative Gromov seminorm and a subsequent infimum step, the paper suggests a two-step strategy for scl calculations in more complex groups and connects these ideas to broader questions about the structure of scl norms and their dual objects. Overall, the results advance understanding of scl and relative seminorms in surface groups, with potential implications for extremal objects and dynamics on surface groups.

Abstract

Let be an injective morphism of free groups. If is geometric (i.e. induced by an inclusion of oriented compact connected surfaces with nonempty boundary), then we show that is an isometric embedding for stable commutator length. More generally, we show that if is a subsurface of an oriented compact (possibly closed) connected surface , and is an integral -chain on , then there is an isometric embedding 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 for a chain in is in fact an admissible surface in .
Paper Structure (25 sections, 32 theorems, 49 equations, 8 figures)

This paper contains 25 sections, 32 theorems, 49 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Scl and the relative Gromov seminorm
  3. Stable commutator length algebraically
  4. Isometric embeddings
  5. Stable commutator length topologically
  6. The Gromov seminorm on homology
  7. The relative Gromov seminorm
  8. Links and orientability in 2-complexes
  9. 2-complexes
  10. A surface criterion for 2-complexes
  11. Orientability of 2-complexes
  12. Subcomplex stability
  13. Standard form for admissible surfaces
  14. Incompressibility and monotonicity
  15. Transversality
  16. ...and 10 more sections

Key Result

Theorem \ref{thm:isom-scl}

Let $S$ be an oriented, compact, connected surface with nonempty boundary and let $T\subseteq S$ be a subsurface. Consider the inclusion-induced map If $\iota_*:H_1\left(\pi_1T\right)\hookrightarrow H_1\left(\pi_1S\right)$ is injective, then $\iota$ is a strong isometric embedding for $\mathop{\mathrm{scl}}\nolimits$.

Figures (8)

  • Figure 1: An inclusion of surfaces that is $H_1$-injective but not isometric for $\mathop{\mathrm{scl}}\nolimits$.
  • Figure 2: Pieces of a transverse incompressible admissible surface.
  • Figure 3: A vertex disc with disconnected link mapping to a vertex in the interior of $S$ (note that the map $f$ is orientation-preserving on the blue cellular disc but orientation-reversing on the green one).
  • Figure 4: Making links of vertex discs connected (interior case).
  • Figure 5: Adjacent cellular discs of opposite orientations.
  • ...and 3 more figures

Theorems & Definitions (66)

  • Theorem \ref{thm:isom-scl}: Isometric embedding for $\mathop{\mathrm{scl}}\nolimits$
  • Example 1.1
  • Theorem \ref{thm:isom-gromnorm}: Isometric embedding for the relative Gromov seminorm
  • Corollary \ref{cor:isom-scl}
  • Definition 2.1
  • Proposition 2.2: Monotonicity of $\mathop{\mathrm{scl}}\nolimits$
  • Definition 2.3
  • Remark 2.4
  • Proposition 2.5
  • proof
  • ...and 56 more