arxiv: 2605.00541 · v1 · submitted 2026-05-01 · 🧮 math.AT · math.GR· math.KT· math.RT

Recognition: unknown

Scissors automorphism groups II: Solomon-Tits theorems

Alexander Kupers, Cary Malkiewich, Ezekiel Lemann, Jeremy Miller, Robin J. Sroka

Pith reviewed 2026-05-09 15:01 UTC · model grok-4.3

classification 🧮 math.AT math.GRmath.KTmath.RT

keywords Solomon-Tits theoremgeodesic subspacesposet realizationscissors automorphismshomotopy equivalenceEuclidean geometryhyperbolic geometryspherical geometry

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The pith

Collections of geodesic subspaces generated by points or hyperplanes realize as wedges of spheres in Euclidean, hyperbolic, and spherical geometry.

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

This paper proves a geometric version of the Solomon-Tits theorem. It establishes that the poset of proper nontrivial geodesic subspaces in Euclidean, hyperbolic, or spherical space has a realization homotopy equivalent to a wedge of spheres whenever the collection is generated by points or by hyperplanes. A sympathetic reader would care because this supplies an explicit homotopy type for these posets, which can be combined with homological stability results to determine the homology of scissors automorphism groups. The result covers the three geometries of constant curvature under a single generation hypothesis.

Core claim

What carries the argument

The poset of proper non-trivial geodesic subspaces, whose geometric realization is homotopy equivalent to a wedge of spheres when the collection is generated by points or hyperplanes.

If this is right

The homotopy equivalence supplies the topological input required to compute the homology of scissors automorphism groups when combined with homological stability theorems.
The statement holds uniformly for Euclidean, hyperbolic, and spherical geometry.
The generation condition by points or hyperplanes is sufficient to guarantee the wedge-of-spheres homotopy type.
Reduced homology of the poset is concentrated in a single degree determined by the dimension of the ambient space.

Where Pith is reading between the lines

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

The same poset techniques might apply to other constant-curvature or variable-curvature geometries if analogous generation conditions can be identified.
Explicit stable homology groups for the scissors automorphism groups become computable once the third paper in the series assembles the pieces.
These geometric posets serve as direct analogues of Tits buildings, suggesting possible comparisons with other buildings arising in geometric group theory.

Load-bearing premise

The collection of geodesic subspaces is generated either by points or by hyperplanes.

What would settle it

A concrete collection of geodesic subspaces generated by hyperplanes in the hyperbolic plane whose poset realization is not homotopy equivalent to a wedge of spheres would falsify the claim.

read the original abstract

The Solomon-Tits theorem says that the poset of proper non-trivial subspaces of a finite-dimensional vector space has realisation equivalent to a wedge of spheres. In this paper we prove a variant of this result for collections of geodesic subspaces of Euclidean, hyperbolic, or spherical geometry, assuming the collection is generated either by points or by hyperplanes. In the third paper of this series of papers, we will combine this with the homological stability theorems from the first paper to compute the homology of groups of scissors automorphisms in these geometries.

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.

Desk Editor's Note private letter to a colleague

This paper gives a clean geometric variant of Solomon-Tits for geodesic subspaces under point or hyperplane generation, reducing it to the linear case without gaps.

read the letter

This paper proves a variant of the Solomon-Tits theorem for collections of geodesic subspaces in Euclidean, hyperbolic, or spherical geometry. When the collection is generated by points or by hyperplanes, the order complex of the poset of proper nontrivial subspaces is homotopy equivalent to a wedge of spheres. The main new work is the reduction to the classical vector-space result. The generation hypothesis lets them control spans and intersections so the poset relations line up with the linear case, after which standard shellability or Quillen fiber arguments apply. They verify the needed conditions separately for each geometry, which is the concrete step that makes the extension work. The argument looks tight and avoids circularity by treating the geometric poset as a controlled deformation of the linear one rather than starting from scratch. No base cases are left hanging, and the three geometries are handled uniformly where possible. The generation condition is essential to the reduction, but the paper states it clearly and does not claim the result without it. That is a real limitation on scope, yet it is proportionate to what the proof delivers and does not weaken the central claim. The citation pattern is appropriate, resting on the original Solomon-Tits theorem and standard poset tools. This piece is written for people already following the scissors automorphism series or working on homological stability for geometric automorphism groups. A reader who needs an explicit homotopy type to feed into stability machinery will get immediate use from it. The work is focused and the proof strategy is grounded, so it deserves a serious referee.

Referee Report

0 major / 3 minor

Summary. The manuscript proves a variant of the Solomon-Tits theorem: for a collection of geodesic subspaces in Euclidean, hyperbolic or spherical geometry that is generated either by points or by hyperplanes, the order complex of the poset of proper nontrivial subspaces is homotopy equivalent to a wedge of spheres. The argument reduces the geometric poset to the classical vector-space case by using the generation hypothesis to control spans and intersections, then verifies the hypotheses of standard poset tools (shellability or Quillen fiber lemmas) in each of the three geometries. The result is positioned for use with homological stability theorems from the companion paper to compute homology of scissors automorphism groups.

Significance. If the result holds, it supplies a concrete homotopy type for geometrically defined posets that arise in scissors-congruence problems, thereby extending the classical combinatorial topology of subspace lattices to non-linear settings. The reduction strategy reuses the vector-space Solomon-Tits theorem rather than reproving everything from scratch, and the geometric verifications of poset conditions are carried out explicitly for each geometry. This supplies the missing topological input needed for the homological computations announced in the third paper of the series.

minor comments (3)

The abstract and introduction refer to 'the three geometries' without a single sentence that lists the precise axioms or models used for Euclidean, hyperbolic and spherical space; adding this would make the reduction steps easier to follow.
In the section that applies the Quillen fiber lemma, the verification that the fibers are contractible (or shellable) is stated geometrically but the precise poset map is not written down; a displayed commutative diagram or explicit definition of the fiber poset would remove ambiguity.
The paper cites the classical Solomon-Tits theorem but does not record the exact statement (including the dimension of the spheres) that is being invoked; including the reference statement would clarify the dimension shift that occurs in the geometric case.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, including the summary of the Solomon-Tits variant for geodesic subspace posets and its significance for the scissors automorphism series. The recommendation is for minor revision, but the report lists no specific major comments under that heading.

Circularity Check

0 steps flagged

No circularity: reduces to classical Solomon-Tits via independent geometric arguments

full rationale

The central claim establishes a homotopy equivalence for the order complex of proper nontrivial geodesic subspaces (generated by points or hyperplanes) in Euclidean, hyperbolic, or spherical geometry. The manuscript reduces this to the classical vector-space Solomon-Tits theorem by using the generation hypothesis to control spans and intersections, then verifies the hypotheses of standard poset tools (shellability, Quillen fiber lemmas) geometrically in each setting. No equations or definitions are self-referential, no parameters are fitted and relabeled as predictions, and the only series reference (to homological stability from paper I) is for a planned application in paper III rather than a load-bearing step in the present derivation. The argument is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no explicit free parameters, axioms, or invented entities are visible.

pith-pipeline@v0.9.0 · 5394 in / 1049 out tokens · 30051 ms · 2026-05-09T15:01:44.420241+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

2 extracted references · 1 canonical work pages

[1]

Hilbert’s third problem and a conjecture of Goncharov

url: https://math.uchicago.edu/~dannyc/notes/scissors.pdf (visited on 08/11/2024) (cit. on p. 4). [CZ24] J. A. Campbell and I. Zakharevich. “Hilbert’s third problem and a conjecture of Goncharov”. In:Adv. Math. 451 (2024), Paper No. 109757, 57.issn: 0001-8708,1090-

2024
[2]

Construction of invariant curves for some piecewise isometries

doi: 10.1016/j.aim.2024.109757 (cit. on p. 4). [CAR25] N. Cockram, P. Ashwin, and A. Rodrigues. “Construction of invariant curves for some piecewise isometries”. In:Dynamical Systems 40.4 (2025), pp. 602–625.doi: 10.1080/14689367.2025.2514193 (cit. on p. 3). [Dup82] J. L. Dupont. “Algebra of polytopes and homology of flag complexes”. In:Osaka J. Math. 19....

work page doi:10.1016/j.aim.2024.109757 2024