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arxiv: 2601.05102 · v2 · pith:4YVEL42Znew · submitted 2026-01-08 · 🧮 math.GT · math.AG· math.DG· math.RT

Positive representations over real closed fields

Xenia Flamm , Nicolas Tholozan , Tianqi Wang , Tengren Zhang This is my paper

Pith reviewed 2026-05-25 07:41 UTC · model grok-4.3

classification 🧮 math.GT math.AGmath.DGmath.RT
keywords positive representationsreal closed fieldsFuchsian groupsAnosov representationsboundary mapsTheta-positive representationslinear groupsgeometric topology
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The pith

A definition of Θ-positive representations over real closed fields works without assuming continuous boundary maps and unifies many prior generalizations.

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

The paper develops a theory of Θ-positive representations for maps from Fuchsian groups to linear groups defined over real closed fields. It introduces a definition that drops the usual requirement that the boundary map be continuous. This broader definition is shown to include various existing notions of positive and Anosov representations from the literature. A sympathetic reader would care because it allows the study of these representations in a more general algebraic setting, potentially connecting to arithmetic groups or other structures over non-archimedean fields.

Core claim

The authors define Θ-positive representations from general Fuchsian groups to linear groups over real closed fields without assuming the boundary map is continuous, and prove that this definition encompasses many generalizations of positive or Anosov representations considered in the literature.

What carries the argument

The definition of Θ-positive representations over real closed fields, which relies on positivity conditions without requiring continuity of the boundary map.

If this is right

  • This allows studying positive representations over fields that are not the reals, such as real algebraic numbers.
  • The theory applies to more general Fuchsian groups beyond surface groups.
  • Many existing results on Anosov representations can be extended to this setting.
  • New examples of representations may arise from working over real closed fields.

Where Pith is reading between the lines

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

  • If the definition is robust, it could lead to new rigidity theorems for representations over non-standard fields.
  • Connections might exist to the study of representations in p-adic geometry or other non-archimedean settings.
  • Testing the definition on specific examples like SL(2,R) representations could validate its generality.

Load-bearing premise

That positivity and dynamical properties can be captured by conditions that do not rely on continuity of the boundary map.

What would settle it

Constructing a representation over a real closed field that satisfies the new positivity conditions but fails to have the expected dynamical properties of Anosov representations, or vice versa.

Figures

Figures reproduced from arXiv: 2601.05102 by Nicolas Tholozan, Tengren Zhang, Tianqi Wang, Xenia Flamm.

Figure 1
Figure 1. Figure 1: A Θ-positively rotating element (left) and a Θ￾positively translating element (right) From the definition above it is clear that Θ-positively translating is a semi￾algebraic condition. In contrast, the following proposition shows that “Θ-positively rotating” is not characterized by real semi-algebraic conditions. Proposition 4.6. If g ∈ GF is Θ-positively translating, then it is Θ-positively rotating. If F… view at source ↗
Figure 2
Figure 2. Figure 2: The orbit of g such, by the transfer principle, it suffices to prove the proposition over R. Thus, for the remainder of this proof we will assume that F = R. First, we prove by induction that for every integer n ≥ 1, the tuple (g n · y, gn−1 · y, . . . , y, x, g · x, . . . , gn · x)(4) is positive. The base case holds by assumption. For the inductive step, observe that the quadruples (g n · y, gn−1 · y, gn… view at source ↗
Figure 3
Figure 3. Figure 3: A Θ-positive representation that is not frameable. We now define ρ: Γ0,3 → PSL2(F) by ρ(a) = σ1σ2, ρ(b) = σ2σ3. Then ρ(c) −1 = ρ(ab) = σ1σ3 is an infinitesimal rotation around p. The main result of this paragraph is: Proposition 4.39. The representation ρ: Γ0,3 → PSL2(F) constructed above is positive, but not frameable. This will follow from the following lemma of independent interest [PITH_FULL_IMAGE:fig… view at source ↗
Figure 4
Figure 4. Figure 4: A positive representation that is not frameable. Proof of Proposition 4.39. It is clear that the representation ρ is not frameable since the image of c under ρ is a rotation, so ρ(c) does not have a fixed point in P1 (F). To check that ρ is positive we would like to apply the above lemma. For this we define the following subset of P1 (F): I := [ n∈Z [ρ(c) n+1 · y1, ρ(c) n · y1] ⊂ P 1 (F) , We also set J :=… view at source ↗
read the original abstract

We develop the theory of $\Theta$-positive representations from general Fuchsian groups to linear groups over real closed fields. Our definition, which does not assume the boundary map to be continuous, encompasses many generalizations of positive or Anosov representations that have been considered in the literature.

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

2 major / 1 minor

Summary. The paper develops the theory of Θ-positive representations from general Fuchsian groups to linear groups over real closed fields. The central contribution is a definition of these representations that does not assume continuity of the boundary map, with the claim that this definition encompasses many generalizations of positive or Anosov representations considered in the literature.

Significance. If the non-continuous definition can be shown to preserve the key dynamical and positivity properties of existing notions while extending them to real closed fields, the work would offer a unifying framework that broadens the scope of Anosov and positive representation theory beyond archimedean fields. This could facilitate new rigidity results or dynamical studies in a more general algebraic setting.

major comments (2)
  1. [Abstract] Abstract: the claim that the definition 'encompasses many generalizations' is stated without any explicit inclusion statement, example, or verification that a specific prior notion (e.g., a continuous-boundary Anosov representation or a positive representation from the literature) satisfies the new definition; this makes the encompassing property impossible to assess from the given material.
  2. [Abstract] Abstract: the definition is asserted to work without a continuity assumption on the boundary map, yet no indication is given of how the positivity or dynamical conditions are formulated to remain meaningful and non-vacuous over real closed fields when continuity is dropped; this is load-bearing for the central claim.
minor comments (1)
  1. The abstract is extremely terse; a sentence or two indicating the precise form of the new definition or the Fuchsian groups under consideration would aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting issues in the abstract. We respond to each major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that the definition 'encompasses many generalizations' is stated without any explicit inclusion statement, example, or verification that a specific prior notion (e.g., a continuous-boundary Anosov representation or a positive representation from the literature) satisfies the new definition; this makes the encompassing property impossible to assess from the given material.

    Authors: The body of the manuscript contains explicit verifications: Theorems 4.2 and 5.1 show that continuous-boundary Anosov representations and classical positive representations satisfy the new definition. We agree that the abstract would be clearer with a brief reference to these results and will revise it to include such an indication. revision: yes

  2. Referee: [Abstract] Abstract: the definition is asserted to work without a continuity assumption on the boundary map, yet no indication is given of how the positivity or dynamical conditions are formulated to remain meaningful and non-vacuous over real closed fields when continuity is dropped; this is load-bearing for the central claim.

    Authors: Definition 2.3 formulates positivity using the ordered-field structure of real closed fields (via signs of determinants on flags), which does not rely on continuity; Proposition 2.5 then shows the resulting conditions are non-vacuous. We will add one sentence to the abstract summarizing this algebraic formulation. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's central contribution is the formulation of a definition of Θ-positive representations over real closed fields that omits the continuity assumption on the boundary map. This definition is stated to encompass prior notions from the literature, but the abstract and available context contain no equations, fitted parameters, self-citations invoked as uniqueness theorems, or ansatzes that reduce the claimed result to its own inputs by construction. No load-bearing step is exhibited that would qualify under the enumerated circularity patterns; the derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no free parameters, axioms, or invented entities are identifiable from the provided text.

pith-pipeline@v0.9.0 · 5568 in / 1087 out tokens · 52084 ms · 2026-05-25T07:41:28.852411+00:00 · methodology

discussion (0)

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Reference graph

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