[Paper Review] Isometric endomorphisms of free groups

Danny Calegari, Alden Walker|arXiv (Cornell University)|Jan 21, 2011

Geometric and Algebraic Topology14 references8 citations

TL;DR

This paper establishes that a random homomorphism between free groups is almost surely an isometry for stable commutator length (scl), demonstrating the existence of abundant exotic isometries in the scl unit ball of free groups. Using fatgraph techniques and vertex quasimorphisms, the authors prove that random elements in free groups with commutator length at most $ n $ have exactly $ \text{scl} = n - \frac{1}{2} $, and provide computable quasimorphisms that certify these results.

ABSTRACT

An arbitrary homomorphism between groups is nonincreasing for stable commutator length, and there are infinitely many (injective) homomorphisms between free groups which strictly decrease the stable commutator length of some elements. However, we show in this paper that a random homomorphism between free groups is almost surely an isometry for stable commutator length for every element; in particular, the unit ball in the scl norm of a free group admits an enormous number of exotic isometries. Using similar methods, we show that a random fatgraph in a free group is extremal (i.e. is an absolute minimizer for relative Gromov norm) for its boundary; this implies, for instance, that a random element of a free group with commutator length at most n has commutator length exactly n and stable commutator length exactly n-1/2. Our methods also let us construct explicit (and computable) quasimorphisms which certify these facts.

Motivation & Objective

• To investigate the prevalence of isometric homomorphisms between free groups for stable commutator length (scl), challenging the expectation that scl is nonincreasing under homomorphisms.
• To demonstrate that random homomorphisms from $ F_k $ to $ F_l $ are isometries for scl with probability $ 1 - O(C^{-n}) $, revealing a rich structure of exotic isometries in the scl unit ball.
• To construct explicit, computable quasimorphisms that certify extremality for random fatgraphs and prove that random elements in free groups with bounded commutator length achieve maximal scl.
• To develop efficient algorithms for verifying extremal quasimorphisms using local conditions and meet-in-the-middle techniques on trivalent fatgraphs.

Proposed method

• A random homomorphism $ \varphi: F_k \to F_l $ is defined by assigning random elements of length at most $ n $ to the generators of $ F_k $, and the probability of being an isometry for scl is analyzed asymptotically.
• The paper uses Generalized Bavard Duality to relate scl to homogeneous quasimorphisms, focusing on vertex quasimorphisms derived from fatgraphs to certify extremality.
• A combinatorial condition (condition B) is introduced to verify whether a labeling of a fatgraph yields a vertex quasimorphism, relying on subword avoidance and small cancellation properties.
• A meet-in-the-middle algorithm is applied to search for valid labelings of trivalent fatgraphs that satisfy the quasimorphism conditions, significantly improving search efficiency.
• Homomorphisms are applied to fatgraphs to transform non-extremal labelings into extremal ones, increasing the success rate of finding extremal quasimorphisms.
• Empirical data from 500,000 random labelings per fatgraph are used to estimate failure rates and fit exponential models for success probabilities.

Experimental results

Research questions

• RQ1 What is the probability that a random homomorphism between free groups preserves stable commutator length?
• RQ2 Can extremal quasimorphisms be efficiently computed for random fatgraphs in free groups?
• RQ3 Do random elements in free groups with commutator length at most $ n $ achieve the maximal possible stable commutator length of $ n - \frac{1}{2} $?
• RQ4 Can the failure rate of finding vertex quasimorphisms be reduced by applying random homomorphisms to the fatgraph?
• RQ5 What combinatorial conditions on fatgraph labelings guarantee the existence of extremal quasimorphisms?

Key findings

• A random homomorphism $ \varphi: F_k \to F_l $ of length $ n $ is an isometry for scl with probability $ 1 - O(C(k,l)^{-n}) $ for some $ C(k,l) > 1 $, proving the existence of exotic isometries in the scl unit ball.
• For a random element in a free group with commutator length at most $ n $, the stable commutator length is exactly $ n - \frac{1}{2} $, and this is certified by extremal fatgraphs.
• The success rate of finding vertex quasimorphisms for trivalent fatgraphs with four vertices is modeled as $ P(\text{success}) \geq 1 - 82.3971(3.19827)^{-n} $, with failure rate decaying exponentially in label length.
• Applying random homomorphisms to fatgraphs increases the success rate of finding extremal quasimorphisms by a factor of about 5, especially for shorter labelings.
• The defect of a vertex quasimorphism can be verified in polynomial time using local subword and constant conditions, unlike general quasimorphisms which require exponential time.
• The Isometry Conjecture posits that any injective homomorphism $ \varphi: F_2 \to F $ is an isometry for scl, supported by extensive computational evidence and theoretical reasoning.