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The Kervaire conjecture and the minimal complexity of surfaces

Lvzhou Chen

TL;DR

The paper develops a novel framework linking the minimal complexity of surfaces bounding a word in a group–via $w$-admissible surfaces in HNN extensions–to nontriviality and injectivity questions for one-relator quotients. By introducing an LP-duality approach with a carefully crafted turn-cost function, the author proves a sharp lower bound $-\chi(S) \ge (1-1/n)\deg(S)$ under $n$-RF conditions, yielding a Freiheitssatz for HNN extensions and new proofs of Klyachko-type injectivity for torsion-free factors. These results imply the Kervaire conjecture in significant cases and establish linear isoperimetric inequalities, hence relative hyperbolicity, for a broad class of one-relator quotients. The work connects stable commutator length techniques to classical one-relator problems, offering new tools and perspectives with potential extensions to more general group decompositions and duality theories.

Abstract

The Kervaire conjecture asserts that adding a generator and then a relator to a nontrivial group always results in a nontrivial group. We introduce new methods from stable commutator length to study this type of problems about nontriviality of one-relator quotients. Roughly, we show that surfaces in certain HNN extensions bounding a given word have complexity no less than the complexity of its boundary. A consequence of this is a Freiheitssatz theorem for HNN extensions, which in particular implies and gives a new proof of Klyachko's theorem that confirms the Kervaire conjecture for torsion-free groups. As another application, we also generalize the following theorem of Klyachko-Lurye to HNN extensions: For any group $G$ and the quotient $Q$ of $G\star\mathbb{Z}$ by any proper power $w^m$ with $w\in G\star\mathbb{Z}$ projecting to $1\in\mathbb{Z}$, the natural map $G\to Q$ is injective.

The Kervaire conjecture and the minimal complexity of surfaces

TL;DR

The paper develops a novel framework linking the minimal complexity of surfaces bounding a word in a group–via -admissible surfaces in HNN extensions–to nontriviality and injectivity questions for one-relator quotients. By introducing an LP-duality approach with a carefully crafted turn-cost function, the author proves a sharp lower bound under -RF conditions, yielding a Freiheitssatz for HNN extensions and new proofs of Klyachko-type injectivity for torsion-free factors. These results imply the Kervaire conjecture in significant cases and establish linear isoperimetric inequalities, hence relative hyperbolicity, for a broad class of one-relator quotients. The work connects stable commutator length techniques to classical one-relator problems, offering new tools and perspectives with potential extensions to more general group decompositions and duality theories.

Abstract

The Kervaire conjecture asserts that adding a generator and then a relator to a nontrivial group always results in a nontrivial group. We introduce new methods from stable commutator length to study this type of problems about nontriviality of one-relator quotients. Roughly, we show that surfaces in certain HNN extensions bounding a given word have complexity no less than the complexity of its boundary. A consequence of this is a Freiheitssatz theorem for HNN extensions, which in particular implies and gives a new proof of Klyachko's theorem that confirms the Kervaire conjecture for torsion-free groups. As another application, we also generalize the following theorem of Klyachko-Lurye to HNN extensions: For any group and the quotient of by any proper power with projecting to , the natural map is injective.
Paper Structure (25 sections, 49 theorems, 67 equations, 6 figures)

This paper contains 25 sections, 49 theorems, 67 equations, 6 figures.

Table of Contents

  1. Introduction
  2. More detailed statements of results
  3. Organization of the paper
  4. Admissible surfaces
  5. A normal form
  6. Basic setup
  7. Putting $S$ in (simple) normal form
  8. The structure of a polygonal boundary
  9. Possible pieces
  10. The gluing graph and Euler characteristic
  11. The LP-duality method
  12. A lower bound of the minimal complexity
  13. The cost function
  14. Comparison with $\chi$
  15. Case I: Only involving $b_i^{\pm1}$'s
  16. ...and 10 more sections

Key Result

Theorem 1

For any torsion-free group $A$, the natural map $A\to H/ \mathopen{\hbox{$\m@th{\langle}$}\hbox{$\m@th{\langle}$}}w \mathclose{\hbox{$\m@th{\rangle}$}\hbox{$\m@th{\rangle}$}}$ induced by the inclusion $A\to H=A\star\mathbb Z$ is injective for any $w\in H$ with $p(w)=\pm 1$, where $p:H\to \mathbb Z$

Figures (6)

  • Figure 1: $X$ has a subspace $X_A$ representing the subgroup $A\le H=\pi_1(X)$ and a loop $\gamma_w$ representing some $w\in H$. $S$ is a connected $w$-admissible surface of degree $4$, where the two boundary components of $S$ on the left are $A$-boundary, mapped to conjugacy classes of $a_1,a_2\in A$, and the three on the right are $w$-boundary of $S$, representing powers of $w$.
  • Figure 2: A boundary-compressible $w$-admissible surface $S$ with $k,m,n\in\mathbb Z_+$ and the simplified $w$-admissible surface $S'=S\setminus \Sigma$, whose boundary representing $w^{n-m}$ (with the orientation induced from $S'$) needs to be further capped off by a disk if $m=n$
  • Figure 3: A $w$-admissible surface $S$ corresponding to an equation of the form (\ref{['eqn: relation']}) in the case $k=4$.
  • Figure 4: $F=f^{-1}(X_C)$ is a set of embedded disjoint proper arcs in the $w$-admissible surface $S$ after applying Lemma \ref{['lemma: no loop']}. After cutting, $S\setminus F$ maps into the thickened vertex space $V$, which deformation retracts to $X_A$.
  • Figure 5: An annulus-piece (left) and a disk-piece (right) glued along paired turns that are of types $(\gamma_i,c,\gamma_j)$ and $(\gamma_{j-1},c^{-1},\gamma_{i+1})$ for some $c\in C$.
  • ...and 1 more figures

Theorems & Definitions (108)

  • Conjecture 1.2
  • Theorem 1: Klyachko, Theorem \ref{['thm: Klyachko']}
  • Conjecture 1.3: Kervaire--Laudenbach
  • Conjecture 1.4: Kervaire
  • Theorem 2: Theorem \ref{['thm: proper power HNN']}
  • Theorem 3: Klyachko--Lurye, Theorem \ref{['thm: proper power']}
  • Conjecture 1.5: Howie Howie_generalize
  • Theorem 4: Corollary \ref{['cor: gap for groups without small torsion']}
  • Theorem 5: Theorem \ref{['thm: HNN general word']}
  • Definition 2.1: $w$-admissible
  • ...and 98 more