Diagram groups and groups of piecewise linear homeomorphisms of the line with global fixed points

Abstract: Assume $n \geq 2$ and $\ell = (r_{1}, \ldots, r_{k}) \in [0,1]<sup>{k}$ is an increasing sequence of real numbers. Let $G_{n,\ell}$ denote the group of orientation-preserving piecewise linear homeomorphisms $h$ of $I = [r_{1}, r_{k}]$ such that: (i) $h&#39;(x)$ is a power of $n$ where it is defined; (ii) if $h&#39;(x)$ is undefined, then $x$ is an $n$-adic rational number, (iii) $h$ fixes each entry of $\ell$, and (iv) $h(\mathbb{Z}[1/n] \cap I) = \mathbb{Z}[1/n] \cap I$. We prove that $G_{n,\ell}$ is a diagram group $D(\mathcal{P}<em>{n,\ell}, ω</em>{n,\ell})$ for all integers $n \geq 2$ and for all finite sequences $\ell$. The semigroup presentation $\mathcal{P}<em>{n,\ell}$ and the word $ω</em>{n,\ell}$ can be computed from the $n$-ary expansions of the numbers $r_{i}$. If all entries in $\ell$ are rational, then $G_{n,\ell}$ has type $F_{\infty}$. Otherwise, $G_{n,\ell}$ is not finitely generated.

• The paper establishes an isomorphism between the groups Gₙ,ℓ and diagram groups, clarifying their algebraic and geometric structures.
• It introduces explicit constructions using n-ary expansions and automata to derive semigroup presentations for PL homeomorphisms.
• Results reveal a dichotomy: groups are of type F∞ when all fixed points are rational, but they fail to be finitely generated when any fixed point is irrational.

Diagram Groups and Groups of Piecewise Linear Homeomorphisms with Fixed Points

Introduction and Motivation

The study of groups of piecewise linear (PL) homeomorphisms on the real line has deep connections with geometric group theory, especially via the generalized Thompson groups. The paper "Diagram groups and groups of piecewise linear homeomorphisms of the line with global fixed points" (2606.27753) focuses on a general class of groups $G_{n,\ell}$, determined by an integer $n \geq 2$ and a finite increasing sequence $\ell = (r_1, ..., r_k)$ in $[0,1]$, where each $G_{n,\ell}$ consists of orientation-preserving PL homeomorphisms of the interval $[r_1, r_k]$ that fix each $r_i\in \ell$, have slopes in the cyclic group generated by $n$, have singularities only at $n$-adic rationals, and permute $n$-adic rationals in the interval.

This paper's principal achievement is to establish an isomorphism between each such $n \geq 2$0 and a diagram group $n \geq 2$1 associated to a computable semigroup presentation, with profound consequences for the group's algebraic and geometric properties. This work both generalizes and systematizes prior results on Thompson's groups and their stabilizers and offers new tools for analyzing finiteness properties via diagrammatic and CAT(0) cube complex methods.

Definition and Structural Analysis of $n \geq 2$2

Given $n \geq 2$3 and $n \geq 2$4 in $n \geq 2$5, the group $n \geq 2$6 is the subgroup of $n \geq 2$7 fixing each $n \geq 2$8 in $n \geq 2$9. Explicitly, $\ell = (r_1, ..., r_k)$0 if and only if:

• $\ell = (r_1, ..., r_k)$1 is a power of $\ell = (r_1, ..., r_k)$2 wherever defined.
• Singularities of $\ell = (r_1, ..., r_k)$3 are located at $\ell = (r_1, ..., r_k)$4-adic rationals in $\ell = (r_1, ..., r_k)$5.
• $\ell = (r_1, ..., r_k)$6.
• $\ell = (r_1, ..., r_k)$7 for all $\ell = (r_1, ..., r_k)$8.

This construction generalizes well-studied objects such as Thompson's group $\ell = (r_1, ..., r_k)$9 (the case $[0,1]$0) and the stabilizer subgroups of finite sets of points.

Realization as Diagram Groups

A central theorem of the paper establishes that $[0,1]$1, with the semigroup presentation $[0,1]$2 and base word $[0,1]$3 computable from the $[0,1]$4-ary expansions of the $[0,1]$5. The translation from the PL data to diagram group data is highly explicit, relying on the construction of automata that encode expansion properties at the breakpoints, and labeling trees that encode permissible subdivisions and local compositions. This isomorphism provides several immediate consequences:

• When all $[0,1]$6 are rational, $[0,1]$7 is finite, and $[0,1]$8 is of type $[0,1]$9 (i.e., it has a classifying space with finite $G_{n,\ell}$0-skeletons for all $G_{n,\ell}$1).
• If any $G_{n,\ell}$2 is irrational, $G_{n,\ell}$3 is not finitely generated—a sharp dichotomy.

The identification with diagram groups allows the direct importation of results on (co)homology (specifically, the free abelian property of integral homology) and properties of group actions on CAT(0) cubical complexes.

Combinatorial and Geometric Constructions

A notable contribution is the explicit combinatorial construction of the semigroup presentation and base word attached to each $G_{n,\ell}$4, involving:

• Automata encoding the $G_{n,\ell}$5-ary expansions of the $G_{n,\ell}$6, taking care to distinguish between periodic and aperiodic expansions and to produce canonical representatives.
• A labeled $G_{n,\ell}$7-ary rooted tree $G_{n,\ell}$8 providing a blueprint for the allowed local generators, partitions, and contraction/expansion moves.
• The development of an associated inverse semigroup $G_{n,\ell}$9 capturing the germ-level data of PL homeomorphisms and allowing for local generation of the group via its action by disjoint partial homeomorphisms.

Additionally, the paper realizes $[r_1, r_k]$0 as acting freely and properly by isometries on a locally finite CAT(0) cubical complex constructed via expansion sets adapted to the combinatorics of $[r_1, r_k]$1 and its higher-dimensional analogs.

Finiteness Properties and Their Proofs

The proof of finiteness properties leverages the established diagram group structure and recent advances in the analysis of expansion sets, as formalized in Farley [Far, J. Group Theory, 2025]. Criteria for being type $[r_1, r_k]$2 are verified by establishing boundedness of contractions, existence of sufficiently many contractions in partitions, contractibility of certain links, and cocompactness/finiteness of stabilizers. Numeric bounds (e.g., $[r_1, r_k]$3 for contraction richness) are made explicit, reflecting the combinatorial complexity of the group in terms of the parameters $[r_1, r_k]$4.

Implications and Outlook

The main results present a robust and extensible framework for understanding a wide class of PL homeomorphism groups with fixed points as diagram groups, unifying diverse strands in the literature on Thompson-like groups, their stabilizers, diagram groups, and geometric group theory. The explicit description of $[r_1, r_k]$5 as a diagram group provides computational and conceptual access to finiteness properties, actions on CAT(0) spaces, and homological invariants.

From a theoretical standpoint, this advances the program of classifying locally defined PL groups via diagrammatic and geometric techniques. The methods used are poised for generalization to broader contexts, such as non-orientation-preserving homeomorphisms or multi-dimensional analogs.

Future work may involve detailed computations of (co)homology, growth types, rigidity phenomena in the space of PL groups, or algorithmic properties associated to the presented semigroup data. Connections to automata theory and formal languages, implicit in the automaton construction for $[r_1, r_k]$6-ary expansions, are also promising for further exploration.

Conclusion

"Diagram groups and groups of piecewise linear homeomorphisms of the line with global fixed points" (2606.27753) introduces an explicit correspondence between a broad class of locally defined PL homeomorphism groups and diagram groups, enabling direct importation of geometric, algebraic, and homological properties. Among the significant results are the concrete criteria for finite presentability (type $[r_1, r_k]$7 in the rational case) and the demonstration that the presence of irrational fixed points precludes finite generation. The technical machinery introduced, particularly the computable passage from fixed point data to diagram group data, is likely to influence further research into the structure and classification of groups arising from PL dynamics and related diagrammatic constructions.