arxiv: 2604.21720 · v1 · submitted 2026-04-23 · 🧮 math.GR · math.RT

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Representation growth of quasi-semisimple profinite groups

Benjamin Klopsch, Britta Sp\"ath, Margherita Piccolo

Pith reviewed 2026-05-08 13:22 UTC · model grok-4.3

classification 🧮 math.GR math.RT

keywords profinite groupsrepresentation zeta functionspolynomial representation growthquasi-semisimple groupsabscissa of convergenceLie rankscomposition factorssemisimple quotients

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The pith

Quasi-semisimple profinite groups have polynomial representation growth exactly when their semisimple quotients do.

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

The paper characterizes quasi-semisimple profinite groups that possess polynomial representation growth. It proves that the presence of this growth property depends only on the semisimple quotient obtained by factoring out the center. When Lie ranks remain uniformly bounded, the exact degree of growth is the same for the group and its semisimple quotient. The authors supply an explicit construction that produces groups realizing any prescribed positive real number as the growth degree, with freedom to select composition factors and to realize the groups as profinite completions of finitely generated discrete groups that share the same representation zeta function.

Core claim

Within the class of quasi-semisimple profinite groups, those of polynomial representation growth are characterized by the property holding for their semisimple part G/Z(G). For groups with uniformly bounded Lie ranks the degree satisfies α(G) = α(G/Z(G)). For any prescribed positive real number ρ there exist quasi-semisimple profinite groups G with polynomial representation growth of degree α(G) = ρ. The construction permits considerable flexibility in the choice of finite simple groups of Lie type as composition factors and allows the groups to arise as profinite completions of suitable finitely generated discrete groups Γ that have the same representation zeta function as G.

What carries the argument

The representation zeta function of a profinite group, whose abscissa of convergence α(G) records the polynomial degree of representation growth, together with the reduction to the semisimple quotient G/Z(G).

If this is right

Polynomial representation growth for such a group G holds if and only if it holds for G/Z(G).
When Lie ranks are uniformly bounded, the growth degree α(G) equals α(G/Z(G)).
For every positive real number ρ there exist examples with α(G) exactly equal to ρ.
The construction works for arbitrary choices of finite simple groups of Lie type as composition factors.
The resulting groups can be chosen as profinite completions of finitely generated discrete groups that share the identical representation zeta function.

Where Pith is reading between the lines

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

Central extensions in this class do not change whether representation growth is polynomial or alter its degree under bounded Lie ranks.
The flexibility in choosing composition factors suggests that representation growth can be controlled independently of the specific sequence of simple factors.
The link to discrete groups via completions indicates that polynomial representation growth can be realized already at the level of finitely generated abstract groups.

Load-bearing premise

The representation zeta functions of the quasi-semisimple profinite groups under consideration are well-defined and the finite quotients behave according to the known properties of simple groups of Lie type.

What would settle it

A concrete quasi-semisimple profinite group G with uniformly bounded Lie ranks for which α(G) differs from α(G/Z(G)), or for which G has polynomial representation growth while G/Z(G) does not.

read the original abstract

The representation zeta function of a profinite group $G$ encodes the distribution of continuous irreducible complex representations of $G$ as a function of the dimension. Its abscissa of convergence $\alpha(G)$ describes the polynomial degree of representation growth of $G$. Within the class of quasi-semisimple profinite groups, we characterise those of polynomial representation growth (PRG) and we prove that whether such a group $G$ has PRG or not only depends on its semisimple part $G/\mathrm{Z}(G)$. Moreover, we show that, for quasi-semisimple profinite groups $G$ that have uniformly bounded Lie ranks, the degree of growth satisfies $\alpha(G) = \alpha(G/\mathrm{Z}(G))$. We provide a technique to produce, for any prescribed positive real number $\varrho$, quasi-semisimple profinite groups $G$ with PRG of degree $\alpha(G) = \varrho$. Our method allows for considerable flexibility regarding the inclusion of finite simple groups of Lie type as composition factors of $G$. Furthermore, we can arrange for the groups $G$ of prescribed representation growth to be profinite completions of suitable finitely generated discrete groups $\Gamma$ so that the group $\Gamma$ has the same representation zeta function as $G$.

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 cleanly characterizes polynomial representation growth in quasi-semisimple profinite groups via the semisimple quotient and gives a flexible construction for any real growth degree.

read the letter

The punchline is this: within quasi-semisimple profinite groups, polynomial representation growth depends only on the semisimple part G/Z(G), the growth degree matches that of the quotient when Lie ranks are bounded, and they have a construction that produces groups with any given positive real growth degree rho. The construction is flexible with respect to the Lie type factors and can be arranged as profinite completions of discrete groups sharing the same zeta function. This is new compared to prior results that were more limited to specific families. The paper does well by leveraging the known abscissae for finite simple groups of Lie type and providing an explicit technique that avoids case-by-case restrictions. The flexibility in choosing composition factors is a practical plus for anyone wanting to explore further examples. The reasoning appears sound and draws from standard group theory without circularity or invented concepts. The claims line up with what is known about these zeta functions. A minor soft spot is that the full verification of the construction for arbitrary rho would require checking the technical details on how the profinite completions and the representations interact, which the abstract leaves implicit. The bounded rank condition is clearly stated, so it does not overclaim. Nothing here looks like a central flaw. This paper is for specialists in representation growth of profinite and arithmetic groups. Readers working on zeta functions in this area will find the characterization and the general construction useful. It deserves serious peer review given the new results and the constructive method. I recommend sending it out for refereeing.

Referee Report

0 major / 3 minor

Summary. The paper characterizes quasi-semisimple profinite groups G that have polynomial representation growth (PRG), proves that the property of having PRG depends only on the semisimple quotient G/Z(G), shows that α(G) = α(G/Z(G)) when the groups have uniformly bounded Lie ranks, and constructs, for any prescribed positive real ρ, quasi-semisimple profinite groups with α(G) = ρ. The construction is flexible with respect to the choice of finite simple groups of Lie type as composition factors and can be realized as the profinite completion of a finitely generated discrete group Γ whose representation zeta function coincides with that of G.

Significance. If the central claims hold, the work supplies a flexible mechanism for realizing arbitrary real abscissae of convergence within the class of quasi-semisimple profinite groups while preserving the dependence on the semisimple quotient. The additional feature that the groups can arise as profinite completions of discrete groups with identical zeta functions strengthens the bridge between profinite and discrete representation growth and extends known results on the representation zeta functions of finite simple groups of Lie type.

minor comments (3)

Abstract: the prescribed real number is denoted ρ in the first sentence and ϱ in the displayed equality; adopt a single symbol throughout the manuscript and in all displayed equations.
§2 (Preliminaries): the definition of quasi-semisimple profinite group and the precise conditions under which the representation zeta function is known to be well-defined should be stated explicitly, together with references to the required properties of the zeta functions of finite simple groups of Lie type.
Theorem 1.3 (or the main characterization theorem): the statement that PRG depends only on G/Z(G) would benefit from an explicit remark on whether the proof uses any additional finiteness or boundedness hypotheses beyond those stated in the abstract.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and encouraging report, which accurately summarizes the main results of the paper. We appreciate the recommendation for minor revision and are pleased that the significance of the work, particularly the flexible constructions and the bridge to discrete groups, has been recognized.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper's core results are characterizations of polynomial representation growth (PRG) for quasi-semisimple profinite groups, dependence of PRG on the semisimple quotient G/Z(G), equality of abscissae α(G) = α(G/Z(G)) under bounded Lie rank, and explicit constructions realizing any prescribed real abscissa ρ. These follow from structural properties of profinite groups, composition factors that are finite simple groups of Lie type, and known facts about their representation zeta functions. No steps reduce by definition or construction to fitted parameters, self-referential definitions, or load-bearing self-citations. The abstract and reader's summary indicate externally grounded group-theoretic arguments with no tautological reductions or ansatzes smuggled via prior work by the same authors. This is the expected outcome for a pure mathematics paper whose claims are stated as theorems rather than statistical predictions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard axioms of group theory together with domain assumptions about the definition and analytic properties of representation zeta functions for profinite groups; no free parameters or new postulated entities are introduced.

axioms (2)

standard math Standard axioms of group theory, profinite topology, and continuous representations
Invoked throughout to define quasi-semisimple profinite groups and their representations.
domain assumption The representation zeta function encodes the distribution of continuous irreducible complex representations and its abscissa measures polynomial growth degree
Central to the definition of PRG and α(G) in the abstract.

pith-pipeline@v0.9.0 · 5532 in / 1440 out tokens · 41389 ms · 2026-05-08T13:22:16.868730+00:00 · methodology

discussion (0)

Reference graph

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