Geometric Inverse Semigroup Theory: a note on the Milnor-Schwarz Lemma for inverse monoids

Abstract: We generalise the Milnor-Schwarz lemma to inverse monoids acting on presheaves of geodesic metric spaces. We provide two proofs of this fact: one only uses elementary techniques, inspired by the arguments for group actions on metric spaces; the other involves a version of the Vietoris-Rips complex, and builds on work of Chung-Martínez-Szakács.

• The paper demonstrates that proper and cobounded actions of inverse monoids on presheaves ensure a finite quasi-generating set and an order-preserving quasi-isometry via the orbit map.
• It employs both a direct analogy with geometric group theory and a Vietoris-Rips complex approach to establish robust geometric frameworks for partial symmetries.
• The study opens avenues for future research on algorithmic properties and large-scale geometry in inverse semigroup theory.

Geometric Structure and Milnor-Schwarz Lemma for Inverse Monoids

Overview

This paper establishes a geometrically robust version of the Milnor-Schwarz lemma for inverse monoids acting on presheaves of geodesic metric spaces, generalizing a foundational result of geometric group theory to the inverse semigroup context. Emphasizing actions via $1$-Lipschitz maps on presheaves structured over the idempotent semilattice $E(S)$ of an inverse monoid $S$, the authors connect algebraic properties (quasi-finite generation) and geometric properties (proper, cobounded actions, and quasi-isometry types), offering two proofs: one by direct analogy with the group-theoretic case, and one via a metric-combinatorial approach using a Vietoris-Rips complex.

Inverse Monoids, Presheaves, and Actions

An inverse monoid $S$ generalizes the notion of group symmetry to partial symmetries, with idempotents corresponding to domains/ranges of partial bijections. The ambient structure $E(S)$ is a meet-semilattice, typically large and often uncountable in nontrivial cases.

The natural geometric context for inverse monoids is their action on presheaves $(X, E(S), p)$ of metric spaces, with the fiber $X_e$ over an idempotent $e \in E(S)$ carrying its own intrinsic metric. Actions preserve fibers (via conjugation at the idempotent level) and are implemented via $1$-Lipschitz maps, essential for metric control in the orbit structure. This construction subsumes the classical group action as a special, degenerate case.

Milnor-Schwarz Lemma for Inverse Monoids

The central theorem (Theorem 1 in the paper) asserts: If $S$ acts properly and coboundedly on a presheaf $E(S)$0 of geodesic metric spaces, then $E(S)$1 admits a finite quasi-generating set, and the orbit map $E(S)$2 (for $E(S)$3) is an order-preserving quasi-isometry from the Cayley metric space $E(S)$4 to $E(S)$5. This is a precise analogue of the classical Milnor-Schwarz lemma for group actions on metric spaces.

Properness is defined via finite control on elements moving a basepoint by bounded distance (modulo cosets of $E(S)$6), while coboundedness requires every point in $E(S)$7 to be approximated uniformly by the action of $E(S)$8 on a base fiber, up to a fixed metric tolerance.

Notably, the result only guarantees quasi-finite generation (existence of a finite generating set modulo the idempotents), not true finite generation, due to the possible complexity and size of $E(S)$9.

Algebraic and Geometric Structures

Key metric objects are Cayley graphs of $S$0 with respect to a quasi-generating set $S$1, interpreted as presheaves of Schützenberger graphs (one per $S$2-class) with the path metric inherited from the underlying digraph. The Cayley metric $S$3 is finite if and only if $S$4 and $S$5 belong to the same $S$6-class, paralleling the group case but adapted for the idempotent structure.

The construction involves precise combinatorial mechanisms to translate algebraic generation by $S$7 and $S$8 into metric properties (geodesic paths) in the Schützenberger graphs. Two generating sets induce biLipschitz equivalent Cayley metrics if both are finite.

Vietoris-Rips Complex and Coarse Geometric Equivalence

The alternative proof leverages the work of Chung-Martínez-Szakács on the coarse geometry of inverse semigroups (2603.29524), introducing a Vietoris-Rips-type graph for the action. This construction yields a uniformly discrete, proper, right-subinvariant extended metric whose connected components are the $S$9-classes of $S$0. The properness and the coarseness type of these metrics characterize quasi-finite generation in inverse semigroups.

These results demonstrate that, under proper and cobounded action, the metric derived from the action (Vietoris-Rips) and the Cayley metric are quasi-isometric, echoing the rigidity and flexibility features seen in the group case.

Theoretical Implications

This extension confirms that inverse monoids acting "geometrically" (i.e., properly and coboundedly) on presheaves of geodesic metric spaces must be quasi-finitely generated, and their large-scale geometry is faithfully recovered by orbit maps, up to quasi-isometry. It recasts many group-theoretic large scale geometry principles—quasi-isometric rigidity, growth, geometric finiteness—in the context of inverse monoids, subject only to the more complex idempotent structure inherent in semigroup theory.

The authors show that no purely local geometric control on action fibers, nor even (hyperbolic) large scale geometry of the fibers, suffices to guarantee finite generation or computational tractability of the word problem. Instead, the presheaf structure and the nature of the restrictions between fibers impose critical constraints.

Open Directions

Several open questions arise: Under what conditions does an inverse monoid act (properly, coboundedly) on presheaves whose fibers satisfy strong geometric constraints (e.g., CAT(0), systolic, or median geometry)? Can the presheaf structure be tuned to enforce better algebraic or algorithmic properties? The connection to algorithmic finiteness and geometric group-theoretic phenomena for inverse semigroups with hyperbolic Schützenberger graphs remains a fertile ground for future investigation.

Conclusion

This paper furnishes a clear geometric framework for inverse monoids, showing that (quasi-)finite generation, metric geometry (through presheaf actions), and coarse equivalence of natural metrics are tightly coupled, thereby providing a solid foundation for further advances in geometric inverse semigroup theory. The methods and perspectives, especially via presheaf and Vietoris-Rips constructions, suggest future pathways connecting large-scale geometry, combinatorial algebra, and the theory of dynamical systems for partial symmetries.