The Wiegold problem and free products of left-orderable groups

TL;DR

The paper resolves the Wiegold problem in a broad setting by proving that free products of nontrivial left-orderable groups have normal rank greater than one, using a novel topological spectral-gap argument for an unsigned stable commutator length and a new construction of relative stackings. The approach hinges on admissible surfaces in a simple normal form and a key inequality relating Euler characteristic to boundary degree, established via relative stackings and dynamical actions on the line. The results yield concrete injections of free factors in quotients by normal closures, provide a rich supply of finitely generated perfect left-orderable groups with high normal rank, and produce 3-manifold applications, including hyperbolic $b Z$-homology spheres that cannot be obtained by Dehn surgery on knots in standard 3-manifolds. The methods employ a versatile blend of topological, dynamical, and combinatorial tools and suggest potential extensions to wider graph-of-groups contexts.

Abstract

A group has normal rank (or weight) greater than one if no single element normally generates the group. The Wiegold problem from 1976 asks about the existence of a finitely generated perfect group of normal rank greater than one. We show that any free product of nontrivial left-orderable groups has normal rank greater than one. This solves the Wiegold problem by taking free products of finitely generated perfect left-orderable groups, a plethora of which are known to exist. We obtain our estimate of normal rank by a topological argument, proving a type of spectral gap property for an unsigned version of stable commutator length. A key ingredient in the proof is an intricate new construction of a family of left-orders on free products of two left-orderable groups.

Figures (9)

• Figure 1: On the left is a $w$-admissible surface $(f,S)$ into a space $X$ with $\pi_1(X)=A\star B$, where $S$ has two boundary components representing $b,b'\in B$, one boundary component representing $a\in A$, and three $w$-boundary components representing $w^k,w^n,w^{-m}$ for some $k,m,n\in\mathbb Z_+$. The subsurface $P$ witnesses its boundary-compressibility, and $S'=S\setminus P$ on the right is the simplified $w$-admissible surface, 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 2: A $w$-admissible surface $S$ corresponding to an equation of the form (\ref{['eqn: group equation']}) with $k=4$.
• Figure 3: The set $J_w=f_w^{-1}(\star)$ on the circle $S^1_w$. The set of disjoint proper arcs $F=f^{-1}(X_C)$ is a set of embedded disjoint proper arcs in a reduced $w$-admissible surface $S$. The $w$-boundary component $C$ has a cover map $p$ to $S^1_w$ that pulls back $J_w$ to $J_C$.
• Figure 4: Compress a $w$-admissible surface $S$ along a loop $L\subset F$ to obtain $S'$. Push the proper arc $\gamma$ away from the disk $D$ bounding $L$ in $S'$ to isotope the pair of pants $P\simeq L_1\cup L_2\cup\gamma$ so that it "lifts" to $S$.
• Figure 5: Left: The proper arcs in $F$ cut a reduced $w$-admissible surface $S$ into $S_A$ and $S_B$, and in this example $S_A$ has two polygonal boundary components and $S_B$ has one. Right: Simplifying $S$ as in Lemma \ref{['lemma: simple normal form']} gives a $w$-admissible $S'$ in simple normal form, which is a union of a disk-piece $P_3$ and two annulus-pieces $P_1$ and $P_2$, where $d(P_1)=d(P_2)=2$, and $d(P_3)=4$.

Theorems & Definitions (69)

• Theorem 1
• Theorem 2
• Corollary 1.1
• Theorem 3
• Theorem 4
• Theorem 5
• Definition 2.1: Relative stacking
• Theorem 6
• Definition 3.1: $w$-admissible
• Remark 3.2