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arxiv: 2204.03278 · v1 · submitted 2022-04-07 · 🧮 math.GR

Finiteness properties of some groups of piecewise projective homeomorphisms

Daniel Farley This is my paper

Pith reviewed 2026-05-24 11:34 UTC · model grok-4.3

classification 🧮 math.GR
keywords Lodha-Moore grouptype F_∞inverse semigrouppiecewise projective homeomorphismsfiniteness propertiesgroup actionslinear fractional transformations
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The pith

The Lodha-Moore group has type F_∞ because it is locally determined by the inverse semigroup S2.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper recasts the Lodha-Moore group G as a group locally determined by the inverse semigroup S2, which is generated by three linear fractional transformations A, B, and C2. Applying a general procedure to this description produces a proof that G has type F_∞. The same argument shows that several other groups acting on the line, the circle, and the Cantor set also have type F_∞. Parallel results are obtained for groups locally determined by the related semigroup S3, which replaces C2 with the translation C3.

Core claim

The Lodha-Moore group G has type F_∞ because it is locally determined by the inverse semigroup S2 generated by the transformations A, B, and C2; the same local-determination property for S3 yields type F_∞ for the corresponding groups, and both cases simultaneously establish type F_∞ for various groups acting on the line, the circle, and the Cantor set.

What carries the argument

The property of being locally determined by an inverse semigroup S2 or S3, which permits direct application of a general procedure to deduce finiteness properties.

If this is right

  • G has type F_∞.
  • Various groups acting on the line, circle, and Cantor set have type F_∞.
  • Groups locally determined by S3 have type F_∞.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The local-determination approach may extend to other piecewise homeomorphism groups generated by similar linear fractional transformations.
  • Type F_∞ for these groups implies they are of type FP_∞ and of type F_n for every finite n.

Load-bearing premise

The groups in question are locally determined by the inverse semigroups S2 and S3.

What would settle it

An explicit computation or counterexample showing that one of the groups locally determined by S2 fails to have type F_∞ would falsify the claim.

Figures

Figures reproduced from arXiv: 2204.03278 by Daniel Farley.

Figure 1
Figure 1. Figure 1: Subdivision trees Remark 6.2. (the subdivision represented by a subdivision tree) Each leaf in a subdivision tree is labelled by a word in the alphabet tA, B, C, cu. The labelling is obtained as follows. Trace the (unique) path p from the root to a given leaf ℓ. Suppose that v1, e1, v2, e2, . . . , ek, vk`1 is a complete list of the vertices and edges encountered along the path p, written in the order that… view at source ↗
Figure 2
Figure 2. Figure 2: The relations that define elementary equivalence be￾tween subdivision trees over S2. n 0 a b c d e 0 0 n+1 0 0 0 -1 -1 1 -1 1 n 0 0 0 e a b c d n-1 0 1 -1 1 -1 1 0 0 [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The relations that define elementary equivalence be￾tween subdivision trees over S3. To apply one of these transformations to a subdivision tree T is to replace a subtree of the form on the left with a subtree of the form on the right. Here the labels a, b, c represent the integer labels of the nodes of T that are attached at the leaves labelled by a, b, c (respectively). An application of the given transf… view at source ↗
Figure 4
Figure 4. Figure 4: Elementary equivalence between two subdivision trees over S2. One easily checks that the two trees are indeed equivalent. Lemma 6.11. If two subdivision trees T1, T2 (over S2 or S3) are elementary equiv￾alent, then they are equivalent. Proof. The proof that the left-hand transformation in [PITH_FULL_IMAGE:figures/full_fig_p019_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Above we have depicted the simplicial complex Epbq associated to b “ ridI , Is by the expansion schemes Ei and E 1 i . An integer k refers to the vertex uk. We recall that rf, ωIs “ rfω, Is when ω P tA, Bu ˚ (see the beginning of Remark 4.8). It follows that the description of Ei is complete for i “ 2 and 3. We next define E 1 i prf,rm, 8qq “ ttrf,rm, 8qsu, trf,rm, m ` 1qs,rf,rm ` 1, 8qsuu. This completes … view at source ↗
Figure 6
Figure 6. Figure 6: The case in which 0 P NpTℓq. a T 1 « T such that npT 1 q “ k ´ 1. However, this implies that k ´ 1 P NpT q, a contradiction. Thus, 0 R NpTℓq, as claimed. We let k1 be either: i) the smallest positive member of NpTℓq, or ii) the largest negative member of NpTℓq. We can assume that npTℓq “ k1 (possibly after replacing Tℓ with an equivalent tree and applying Lemma 6.6). We note that k1 ´ 1 R NpTℓq (in case i)… view at source ↗
read the original abstract

The Lodha-Moore group $G$ first arose as a finitely presented counterexample to von Neumann's conjecture. The group $G$ acts on the unit interval via piecewise projective homemorphisms. A result of Lodha shows that $G$ in fact has type $F_{\infty}$. Here we will describe $G$ as a group that is "locally determined" by an inverse semigroup $S_{2}$, in the sense of the author's joint work with Hughes. The semigroup $S_{2}$ is generated by three linear fractional transformations $A$, $B$, and $C_{2}$, where $A$ and $B$ are elliptical transformations of the hyperbolic plane and $C_{2}$ is a hyperbolic translation. Following a general procedure delineated by Farley and Hughes, we offer a new proof that $G$ has type $F_{\infty}$. Our proof simultaneously shows that various groups acting on the line, the circle, and the Cantor set have type $F_{\infty}$. We also prove analogous results for the groups that are locally determined by an inverse semigroup $S_{3}$, which shares the generators $A$ and $B$ with $S_{2}$, but replaces $C_{2}$ with a different hyperbolic translation $C_{3}$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript claims that the Lodha-Moore group G, acting on the unit interval by piecewise projective homeomorphisms, is locally determined by the inverse semigroup S2 generated by the linear fractional transformations A, B, and C2. Invoking the general Farley-Hughes finiteness criterion then yields a new proof that G has type F_∞. The same argument simultaneously establishes type F_∞ for various groups acting on the line, the circle, and the Cantor set, and analogous results are proved for the groups locally determined by the inverse semigroup S3 (which replaces C2 by the hyperbolic translation C3).

Significance. If the local-determination statements are verified, the work supplies an alternative proof of a known result on G together with extensions to related actions and to the S3 case. The simultaneous treatment of multiple spaces and the explicit reduction to the existing Farley-Hughes machinery constitute the main strengths; the paper thereby illustrates the reach of the inverse-semigroup framework for finiteness properties of homeomorphism groups.

minor comments (3)
  1. [Abstract] The abstract refers to 'various groups acting on the line, the circle, and the Cantor set' without naming them; a parenthetical list or reference to the relevant sections would improve immediate readability.
  2. [Introduction] The introduction would benefit from a short, self-contained statement of the precise generators and the local-determination condition for S2 before the appeal to the general procedure is made.
  3. Notation for the actions on the line, circle, and Cantor set is introduced only in the later sections; a uniform notation table or diagram early in the paper would aid comparison across the simultaneous results.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive report, accurate summary of the manuscript, and recommendation to accept. We are pleased that the simultaneous treatment of multiple spaces and the reduction to the Farley-Hughes criterion were viewed as strengths.

Circularity Check

0 steps flagged

No significant circularity; derivation applies external general theorem to new local-determination claim

full rationale

The paper verifies that G (and variants) is locally determined by the inverse semigroup S2 (resp. S3) generated by the listed linear fractional transformations, then invokes the general Farley-Hughes finiteness criterion from prior joint work. This verification step is presented as new content specific to the groups in question, and the general procedure is treated as an independent theorem rather than redefined or fitted here. No equation or claim reduces the type F_∞ conclusion to a self-referential definition, a renamed empirical pattern, or a self-citation chain whose validity depends on the present manuscript. The self-citation is therefore non-circular support for the application.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract, the central claim rests on the applicability of the general procedure from prior joint work with Hughes to the specific semigroups S2 and S3; no free parameters, invented entities, or ad-hoc axioms are mentioned.

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Works this paper leans on

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