Factorization Semigroups in Algebra & Geometry

• Factorization semigroups are algebraic structures that encode group factorizations using Hurwitz moves, formalizing the passage from local arrangements to global invariants.
• They bridge combinatorial data and geometric applications by utilizing homomorphisms to track global monodromy and ramification profiles in coverings and Hurwitz spaces.
• Finitely generated and algorithmically tractable, these semigroups enable complete classifications of covering spaces, as illustrated by the explicit S3 example.

A factorization semigroup is an algebraic structure encoding the combinatorics of factorizations of elements in a group or monoid, refined by equivalence moves such as Hurwitz moves or other relations, and is intimately connected to both classical and modern theories in algebra, geometry, and combinatorics. Its paper formalizes the passage from local factorizations to global invariants, such as parametrizations of Hurwitz spaces, classification of covering spaces, and the structure theory of monodromy representations. The semigroup structure captures not only the set-data of factorization configurations but their concatenation and transformation rules, providing a combinatorial framework for organizing both algebraic and geometric information.

1. Algebraic Construction and Presentation

Let $(G, O)$ be a finite group equipped with a conjugation-invariant subset $O$ (e.g., $O$ is a union of conjugacy classes in $G$). The factorization semigroup $S(G, O)$ is the quotient of the free semigroup on the alphabet $\{x_g : g \in O\}$ by a system of Hurwitz-type relations

$x_g \cdot x_h = x_{g h g^{-1}} \cdot x_g$

for all $g, h \in O$. This relation reflects the fundamental transformation in the configuration of factors induced by exchanging neighboring elements with a twist by conjugation.

The semigroup thus encodes:

• The ordered collection of factors, each from $O$.
• Equivalence under elementary Hurwitz moves (permutations via conjugation).
• A concatenation law inherited from the underlying free semigroup.

Further algebraic structure arises via:

• The product homomorphism $a: S(G, O) \to G$, with $a(x_g) = g$, tracking the total group element represented by a word.
• The type homomorphism $T: S(G, O) \to \mathbb{Z}_+^m$, recording, for a fixed decomposition of $O$ into $m$ conjugacy classes, the number of times each class appears in the factorization.

Algebraic stability of $S(G, O)$ plays a crucial role: there exists a stabilizing element $s_0$ such that multiplying any two elements of the same type and same product in $G$ by $s_0$ makes them equal in the semigroup. Central elements, such as certain Hurwitz elements in the semigroup over $S_d$, underpin this stabilization property.

2. Relationship with Hurwitz Spaces and Geometric Applications

Classically, the (marked) Hurwitz space $\mathrm{HUR}_\alpha$ parameterizes degree $d$ branched covers of $\mathbb{P}^1$ with prescribed ramification types. The monodromy tuple $(g_1,\ldots,g_b) \in O^b$ encodes the local monodromies at each branch point, required to multiply to the identity (or a specified global monodromy element). Equivalence is given by the Hurwitz action of the braid group, exactly encapsulated by the Hurwitz relations in $S(G, O)$.

The main consequences are:

• There is a canonical identification between irreducible components of the Hurwitz space and equivalence classes (under Hurwitz moves) of such monodromy factorizations, i.e., elements of the relevant subsemigroup of $S(G, O)$.
• For fixed branch data (ramification types), the collection of irreducible components correspond precisely to words in $S(G, O)$ of specified length.
• When the type (ramification profile) is sufficiently 'large'—for instance, involving at least $3(d-1)$ transpositions for degree $d$ covers—the global monodromy is uniquely pinned down by the local data (type), and thus the component of the Hurwitz space is uniquely determined.

Thus, the factorization semigroup not only organizes the arithmetic and combinatorics of factorizations but provides a coordinate system for the decomposition of Hurwitz spaces.

3. Structural Properties and Finiteness

The factorization semigroup $S(G, O)$ is finitely generated and finitely presented. The generators correspond to the distinct elements of $O$ (or, more efficiently, to representatives for the conjugacy classes comprising $O$). The defining relations are all consequences of the universal lift of Hurwitz moves to the semigroup context.

For practical computation and classification, this implies:

• $S(G, O)$ can be algorithmically handled, at least in principle, using combinatorial group-theoretic methods.
• The classification problem for monodromy factorizations, up to Hurwitz equivalence, reduces to the word problem in this semigroup.
• Central elements and stabilization properties further simplify the parametrizations and allow in many cases for reduction to normal forms.

4. Explicit Example: Factorization Semigroup for $S_3$ and Complete Classification

The paper provides the concrete paper of $S(G, O)$ when $G = S_3$ and $O$ is the union of transpositions and 3-cycles. The semigroup $\Sigma_{3,1}$ in this case is generated by

$$x_{(1,2)},\; x_{(1,3)},\; x_{(2,3)},\; x_{(1,2,3)},\; x_{(1,3,2)}$$

subject to specific relations mirroring the braid action and reflecting all admissible Hurwitz moves (relations (28)-(34) in the paper).

Key features:

• Presentations: E.g., $S_i \cdot S_j = S_j \cdot S_i$ for all $i, j$, $S_1 \cdot S_2 = S_1 \cdot S_3 = S_2 \cdot S_3$, and relations connecting other products.
• Every element in $\Sigma_{3,1}$ (i.e., every monodromy factorization representing a three-sheeted cover under prescribed ramification types) admits a unique normal form up to simultaneous conjugation.
• The irreducible components of the Hurwitz space for three-sheeted covers of $\mathbb{P}^1$, $\mathrm{HUR}_3(\mathbb{P}^1)$, correspond bijectively to classes in the semigroup. In the case of global trivial monodromy, the number of irreducible components is explicitly computable ([6] + 1 in the normalization used by the authors).

This explicit presentation demonstrates both the practicality of the semigroup encoding and its power to completely characterize the geometry of Hurwitz spaces in low degree.

5. Types, Homomorphisms, and Rigidity Phenomena

The homomorphisms $a: S(G, O) \to G$ and $T: S(G, O) \to \mathbb{Z}_+^m$ provide the principal means of connecting the semigroup structure with arithmetic invariants and moduli-theoretic parameters:

• The product homomorphism $a$ tracks the global monodromy (total covering monodromy permutation), determining the Galois type.
• The type homomorphism $T$ records the ramification profile—how many branch points of each type occur.

A significant outcome is that, in the regime of 'large' or 'stable' types, the set of local ramification types determines the global object uniquely (rigidity), reflected algebraically in the semigroup via stabilization.

6. Broader Significance and Connections

Factorization semigroups serve as a global combinatorial 'envelope' for factorizations in finite groups, with direct implications for:

• The paper of Galois covers and invariants of their moduli.
• The structure theory of Hurwitz spaces, especially the decomposition into irreducible components in relation to prescribed local data.
• The connection between braid group actions, monodromy invariants, and algebraic presentations.
• Algorithmic aspects of classifying and enumerating covers, which reduce to semigroup-theoretic computations.

In higher degree or for more general groups, stabilization, central elements, and homomorphisms $a, T$ remain critical tools for classification and for understanding when local-to-global rigidity holds.

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In summary, factorization semigroups offer a rigorous, bridge-building framework that translates between group-theoretic factorizations (augmented by Hurwitz equivalence) and the geometry of families of coverings, with structure that enables both theoretical insights and effective computation of moduli and classification results (Kulikov, 2010).