Which reducible representations are Anosov?

Max Lahn

Which reducible representations are Anosov?

T0 review · 0 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A block-diagonal representation is Anosov exactly when there is a unique integer table Q of block eigenvalue counts that makes the cross-block eigenvalue ratios grow linearly.

desk verdict A solid, self-contained characterization of Anosov representations of block diagonal form, with correct deformation-theoretic consequences; the main theorems are new and the proofs hold up. read the letter →

arxiv 2411.15321 v1 pith:LAPDKEYN submitted 2024-11-22 math.GR math.GT

Max Lahn This is my paper · ORCID

classification math.GRmath.GT MSC 20F6722E40

keywords Anosovrepresentationsreducibleeigenvaluegapsblockdiagonalcharactervarietyhyperbolicgroupslargeconfigurationlimitmaps

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper gives a complete characterization of when a reducible representation of a non-elementary word hyperbolic group is Anosov, in terms of the eigenvalue magnitudes of its irreducible blocks. It shows that a block diagonal representation is P_theta-Anosov precisely when one can assign, to each block and each flag level, a nonnegative integer q_{j,k} so that the q_{j,k}-th eigenvalue of one block compared with the (q_{j,k}+1)-th eigenvalue of another block grows at least linearly along the group. This reduces the Anosov property to a check of finitely many eigenvalue ratios, and it determines the Anosov limit map completely from the integer configuration and the block-level limit maps. As a consequence, the deformation spaces of reducible Anosov representations are convex, bounded, open sets, and for many hyperbolic groups, connected components of the character variety that consist entirely of Anosov representations cannot contain reducible ones.

What carries the argument

The central object is the large eigenvalue $\theta$-configuration Q = (q_{j,k}), a family of integers indexed by the block number j and the flag level k, with 0 <= q_{j,k} <= dim_K(U_j) and sum_j q_{j,k} = k. It records how many of the k largest eigenvalue magnitudes of the full representation are contributed by the j-th block. The load-bearing identity is the eigenvalue-gap formula of Corollary 3.3: log( lambda_k(rho(gamma)) / lambda_{k+1}(rho(gamma)) ) = min_{i,j} log( lambda_{q_{i,k}}(eta_i(gamma)) / lambda_{q_{j,k}+1}(eta_j(gamma)) ), which converts the Anosov linear-growth condition into cross-block eigenvalue ratios. The structured-flag spaces W_{Q}^{U,$\theta$}(V) and S_{Q}^{U,$\theta$}(V) then encode exactly which flags can occur as limit flags, namely those meeting each block in dimension q_{j,k}.

What would settle it

Take a non-elementary word hyperbolic group such as a free group F_2, a direct sum V = U_1 oplus U_2, and a block diagonal P_theta-Anosov representation with both blocks irreducible; compute the limit flag at two different boundary points and check whether dim(U_j cap xi_k(z)) is constant in z and equals the unique admissible integer q_{j,k}. If the dimension varies with z, or if two different admissible families both satisfy the linear-growth condition for all infinite-order elements, then Theorem A is false.

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Extended reading notes

Core claim

The central claim is Theorem 3.4 (and its block-diagonal specialization, Theorem A): if a representation rho is block upper triangular relative to a direct sum decomposition U, with block diagonalization eta = oplus_{j=1}^m eta_j, then rho is P_theta-Anosov if and only if there is a unique (U,$\theta$)-admissible family Q = (q_{j,k}) of integers 0 <= q_{j,k} <= dim_K(U_j) such that log( lambda_{q_{i,k}}(eta_i(gamma)) / lambda_{q_{j,k}+1}(eta_j(gamma)) ) grows at least linearly in translation length for all admissible i,j,k. In that case Q is the large eigenvalue $\theta$-configuration: for every boundary point z, q_{j,k} = dim_K( U_j cap xi^k_eta(z) ), each block eta_j is P_{theta_j}-Anosov with limit map obtained by intersecting the full limit flag with U_j, and the full limit flag decomposes as the direct sum of the contributing block limit flags. This reduces the Anosov condition for reducible representations to eigenvalue gaps of the blocks, and it shows that the discrete configuration Q is a genuine invariant of the representation.

Load-bearing premise

The argument needs the group's Gromov boundary to be rich enough that the orbit of any hyperbolic element's attracting point is dense; for elementary groups like Z the configuration Q can depend on the element, so the characterization fails without the non-elementary hypothesis.

Editorial extensions

If this is right

For any reducible representation, checking the Anosov property is reduced to checking linear growth of finitely many cross-block eigenvalue ratios, once the integer table Q is known.
The Anosov limit map of a block diagonal representation is explicitly computable from the block limit maps via the direct-sum formula xi_eta(z) = ( oplus_{q_{i,k}>0} xi^{q_{i,k}}_{eta_i}(z) )_k, so the configuration Q determines the entire asymptotic flag.
The space A^U_theta(zeta) of block deformations that preserve the P_theta-Anosov property is always convex and bounded (Theorem C), giving a constrained deformation theory for reducible Anosov representations.
If the commutator subgroup [Gamma,Gamma] has infinite index in Gamma, then any connected component of the character variety consisting entirely of Anosov representations contains no reducible representation (Corollary D).
Each block eta_j with theta_j nonempty is itself P_{theta_j}-Anosov, so the Anosov property propagates down the block decomposition to the irreducible factors.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The uniqueness of Q suggests a natural stratification of the reducible Anosov locus by the discrete data Q, which could lead to a cell decomposition of the corresponding character variety components; the paper does not develop this.
The Z counterexample shows the non-elementary hypothesis is not merely technical: for elementary groups the large-eigenvalue configuration can depend on the chosen hyperbolic element, so extending the result would require a different invariant.
Theorem B's supremum over infinitely many group elements may be computable by testing only primitive elements in the torsion-free abelianization, as the paper notes, and the paper's Question 4.3 asks whether the deformation domains are finite-sided polytopes; this could be tested on explicit free-group or surface-group examples.
One could in principle use the characterization as an algorithm: enumerate admissible Q, check the cross-block eigenvalue ratios on a generating set, and thereby decide whether a given reducible representation is Anosov for hyperbolic groups with manageable geometry.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

You should know this is a good paper. The main result (Theorem A/3.4) reduces the Anosov condition for block diagonal representations to an eigenvalue-gap condition between blocks, governed by a unique combinatorial object Q, the large eigenvalue configuration. This is genuinely new: it generalizes the author's earlier q-Fuchsian suspension result to all reducible representations and does not reduce to prior work. The proof is long but coherent: a generalized-eigenvector analysis of block upper triangular maps, followed by a clean eigenvalue-gap argument for the converse. I looked at the non-elementarity hypothesis. It is really needed, because the density of attracting fixed point orbits in the Gromov boundary fails for elementary groups, and the paper explicitly gives the Z counterexample. So that is not a flaw.

The deformation-theoretic part is also solid. Theorem C says the Anosov deformation domains are convex and bounded open subsets; the boundedness proof has a potential pitfall with rays, but it works because if a homomorphism to R is positive on one block at a group element, it is negative on that block at the inverse. Corollary D, about Anosov-only components avoiding reducibles when the commutator has infinite index, is a natural and correct consequence. The paper is careful about the character variety quotient and about eigenvalue magnitude continuity. Citation practice is fine: the author cites their previous work for context, but the central theorems are proved from standard background (KP22, GW12, Gue+17). The only blemish I found is minor: in Theorem 3.4(3), the notation xi^{q_i,k}_{eta_i}(z) is used when q_i,k = dim_K(U_i), just outside the usual range of the P_{theta_i}-Anosov limit map. It is harmless and can be fixed by declaring that flag component to be the whole space. There are a few typos and some implicit standard facts about hyperbolic groups, but nothing that affects correctness.

This paper is for anyone working on Anosov representations, higher Teichmuller theory, or character varieties of hyperbolic groups. It deserves a serious referee. I would accept it with minor revisions, asking for the technical fix in 3.4(3) and a remark on the finite-order case in the eigenvalue gap formula. Verdict: accept.

Referee Report

0 major / 5 minor

Summary. The paper studies when reducible representations of a non-elementary word hyperbolic group into GL(V) are Anosov. The main result, Theorem A (proved as Theorem 3.4), characterizes P_theta-Anosov block upper triangular representations in terms of the eigenvalue magnitudes of the blocks of their block diagonalization, via a uniquely defined admissible family Q = (q_{j,k}) of dimensions describing how the top k eigenvalues are distributed among the blocks. The paper also proves that the space of block deformations yielding Anosov representations is a bounded convex open set (Theorem C), gives a concrete description of this space when the block normalization is itself Anosov (Theorem B), and derives a corollary about connected components of character varieties containing reducible representations (Corollary D). The proofs are detailed and largely self-contained, building on a generalized-eigenvector analysis of block upper triangular matrices (Proposition 3.1) and the Kassel--Potrie eigenvalue-gap characterization of Anosov representations.

Significance. If the characterization holds, it provides a complete and explicit criterion for Anosov reducible representations, reducing the problem to exponential growth rates of eigenvalue gaps between blocks. This is a natural and useful generalization of the author's earlier work on reducible suspensions, and the convexity/boundedness theorems give a clean deformation-theoretic picture. The paper is careful with the non-elementary hypothesis, including a counterexample showing that the uniqueness of the configuration Q fails for Z. The proofs are thorough and, apart from local presentation issues, the central claims are well supported.

minor comments (5)

[Lemma 4.2] Lemma 4.2 states that [Gamma,Gamma] has finite index if and only if hom(Gamma,D_U)={0}. This is false as stated for the trivial decomposition U=(V), for which D_U={0} while [Gamma,Gamma] may have infinite index (e.g., a free group). The proof already uses dim(D_U)>0, so the statement should assume U is non-trivial or at least that dim(D_U)>0. The false direction is not used in Corollary D, so this is a local issue.
[Theorem 3.4(3)] In the formula for the limit map xi^theta_eta(z), the notation xi^{q_{i,k}}_{eta_i}(z) is used for q_{i,k}=dim_K(U_i), but the P_{theta_i}-Anosov limit map is only defined for indices in Delta_{U_i}. This is harmless if one adopts the convention that the full-block component is the whole space U_i, but the convention should be stated explicitly.
[Corollary 3.3, proof of (3)] In the proof, the indexing condition for the maximum is written as 'q_{j,k}<dim_K(U'_j cap A^+_k)', which should be 'q_{j,k}<dim_K(U_j)'. The displayed formula in the statement of Corollary 3.3(3) already uses the correct condition, so this is a typo in the proof.
[Theorem C, boundedness proof] The line 'This contradiction implies that psi_i(gamma)<psi_j(gamma) for some j and gamma' is logically too quick. The contradiction only rules out psi_i=psi_j for all j and all gamma. To obtain the strict inequality, one uses that psi_i and psi_j are homomorphisms: if psi_i(gamma)>psi_j(gamma), then at gamma^{-1} the inequality reverses. Adding this observation makes the argument complete.
[Throughout] There are several minor typographical errors, including 'representions' in the abstract and an 'upslope' artifact in the proof of Lemma 4.2. These should be corrected in a final pass.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Anosov characterization and deformation-space results are derived from external eigenvalue-gap criteria and internal constructions, with self-citations only motivational.

full rationale

The derivation chain is self-contained. Theorem 3.4 takes as input the external Kassel–Potrie characterization [KP22, Corollary 4.6], adopted as the definition of Pθ-Anosov, and proves the block-diagonal characterization from scratch: the forward direction constructs the large eigenvalue θ-configuration Q from the attracting flag of a fixed infinite-order element, uses density of the Γ-orbit of the attracting point in ∂Γ (valid because Γ is non-elementary, with the Z counterexample explicitly acknowledged) to show Q is a well-defined invariant of the limit map, and then derives the gap formula from Corollary 3.3; the converse direction only needs admissibility of Q and the strict eigenvalue ordering to compare λ_k and λ_{k+1}, not the target conclusion. The uniqueness of Q follows from the same limit-map construction. The deformation theorems are also proved internally: Theorem C uses Lemma 4.1's convexity computation and a ray-contradiction argument for boundedness, and Theorem B derives the sup-characterization from Theorem 3.4 and the fact that a commutator-subgroup element kills δ and φ. Citations to [Lah24] occur only as motivation or analogy, for example 'in analogy to [Lah24, Theorem 2]' and 'in analogy to [Lah24, Theorem 1]', and do no logical work in the proofs. No equation in the paper presumes its own conclusion, and no fitted parameter is renamed as a prediction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 2 invented entities

The central results rest entirely on standard background theorems in hyperbolic groups and Anosov representation theory; the paper introduces no fitted constants. The new combinatorial definitions (large eigenvalue configuration, structured flags) are internal tools proven in the text.

assumptions (4)

standard math Kassel-Potrie eigenvalue gap characterization of P_theta-Anosov representations ([KP22, Corollary 4.6])
Used as the definition of Anosov throughout (Section 2.2); the paper does not reprove it.
standard math Openness/stability of the Anosov condition in representation and character varieties ([GW12, Theorem 1.2])
Invoked to show A^U_theta(zeta) is open (Theorem C) and to rule out the s=1 boundary in Theorem B (Section 4).
standard math For a non-elementary hyperbolic group, the orbit of the attracting fixed point of any hyperbolic element is dense in the Gromov boundary; and every non-elementary hyperbolic group contains a non-abelian free subgroup
The density fact is used in Theorem 3.4 to show Q is independent of the group element; the free subgroup fact guarantees [Gamma,Gamma] contains an infinite order element in Theorem C.
domain assumption Semisimplification invariance: a representation is Anosov if and only if its block diagonalization/semisimplification is ([Gue+17, Section 2.5.4], reproved in Corollary 2.10)
This reduction from reducible to block diagonal representations is the entry point of the whole paper.

invented entities (2)

large eigenvalue theta-configuration Q
purpose: Records, for each block U_j and each k in theta, the number of large eigenvalue directions of the block diagonal representation that lie in U_j; it determines the limit map formulas and the deformation domain.
Defined in Section 3.1 and proven to exist uniquely for Anosov block diagonal representations (Theorem A); no external experimental handle.
weakly and strongly (U,theta,Q)-structured flags
purpose: Subsets of the flag variety that contain the limit flags of Anosov block diagonal representations; used to state Theorem 3.4(1).
Internal definitions; their role is established by Theorem 3.4, with no independent external evidence.

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Cite this review

Pith. "Pith review of Which reducible representations are Anosov?." pith.science (2026). https://pith.science/paper/LAPDKEYN

@misc{pith2026241115321,
  author       = {Pith},
  title        = {Pith review of: Which reducible representations are Anosov?},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LAPDKEYN}},
  note         = {Machine review of arXiv:2411.15321}
}

read the original abstract

We give a characterization of the Anosov condition for reducible representations in terms of the eigenvalue magnitudes of the irreducible block factors of its block diagonalization. As in previous work, these Anosov representations comprise a collection of bounded convex domains in a finite-dimensional vector space, and this perspective allows us to conclude for many non-elementary hyperbolic groups that connected components of the character variety which consist entirely of Anosov representations do not contain reducible representations.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

3 extracted references · 1 canonical work pages

[1]

Three-dimensional Anosov ﬂag man ifolds

[Bar10] Thierry Barbot. “Three-dimensional Anosov ﬂag man ifolds”. In: Geometry & Topology 14.1 (Jan. 2010), pp. 153–191. doi: 10.2140/gt.2010.14.153. [Bri+15] Martin Bridgeman et al. “The pressure metric for An osov representations”. In: Geometric and Functional Analysis 25.4 (June 2015). doi: 10.1007/s00039-015-0333-8 . [Gué+17] François Guéritaud et al...

work page doi:10.2140/gt.2010.14.153 2010
[2024]

Department of Mathematics, University of Michigan, Ann Arb or, MI 48109 Email address : maxlahn@umich.edu

doi: 10.48550/arXiv.2312.09886. Department of Mathematics, University of Michigan, Ann Arb or, MI 48109 Email address : maxlahn@umich.edu

arXiv doi:10.48550/arxiv.2312.09886
[3923]

Eigenvalue gaps for hyperbolic groups and semigroups

doi: 10.2140/gt.2018.22.3827. [KP22] Fanny Kassel and Rafael Potrie. “Eigenvalue gaps for hyperbolic groups and semigroups”. In: Journal of Modern Dynamics 18 (2022), pp. 161–208. doi: 10.3934/jmd.2022008. [Lab06] François Labourie. “Anosov ﬂows, surface groups an d curves in projective space”. In: Inventiones mathematicae 165.1 (Mar. 2006), pp. 51–114. d...

doi:10.2140/gt.2018.22.3827 2022

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Three-dimensional Anosov ﬂag man ifolds

[Bar10] Thierry Barbot. “Three-dimensional Anosov ﬂag man ifolds”. In: Geometry & Topology 14.1 (Jan. 2010), pp. 153–191. doi: 10.2140/gt.2010.14.153. [Bri+15] Martin Bridgeman et al. “The pressure metric for An osov representations”. In: Geometric and Functional Analysis 25.4 (June 2015). doi: 10.1007/s00039-015-0333-8 . [Gué+17] François Guéritaud et al...

work page doi:10.2140/gt.2010.14.153 2010

[2] [2024]

Department of Mathematics, University of Michigan, Ann Arb or, MI 48109 Email address : maxlahn@umich.edu

doi: 10.48550/arXiv.2312.09886. Department of Mathematics, University of Michigan, Ann Arb or, MI 48109 Email address : maxlahn@umich.edu

arXiv doi:10.48550/arxiv.2312.09886

[3] [3923]

Eigenvalue gaps for hyperbolic groups and semigroups

doi: 10.2140/gt.2018.22.3827. [KP22] Fanny Kassel and Rafael Potrie. “Eigenvalue gaps for hyperbolic groups and semigroups”. In: Journal of Modern Dynamics 18 (2022), pp. 161–208. doi: 10.3934/jmd.2022008. [Lab06] François Labourie. “Anosov ﬂows, surface groups an d curves in projective space”. In: Inventiones mathematicae 165.1 (Mar. 2006), pp. 51–114. d...

doi:10.2140/gt.2018.22.3827 2022