Representation growth of quasi-semisimple profinite groups
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| Format: | Preprint |
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2026
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| author | Klopsch, Benjamin Piccolo, Margherita Späth, Britta |
| author_facet | Klopsch, Benjamin Piccolo, Margherita Späth, Britta |
| contents | 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 $α(G)$ 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 $α(G) = α(G/\mathrm{Z}(G))$. We provide a technique to produce, for any prescribed positive real number $\varrho$, quasi-semisimple profinite groups $G$ with PRG of degree $α(G) = \varrho$. 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 $Γ$ so that the group $Γ$ has the same representation zeta function as $G$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_21720 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Representation growth of quasi-semisimple profinite groups Klopsch, Benjamin Piccolo, Margherita Späth, Britta Group Theory Representation Theory Primary 20E18, Secondary 11M41, 20C15, 20C33, 20D06, 20F69 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 $α(G)$ 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 $α(G) = α(G/\mathrm{Z}(G))$. We provide a technique to produce, for any prescribed positive real number $\varrho$, quasi-semisimple profinite groups $G$ with PRG of degree $α(G) = \varrho$. 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 $Γ$ so that the group $Γ$ has the same representation zeta function as $G$. |
| title | Representation growth of quasi-semisimple profinite groups |
| topic | Group Theory Representation Theory Primary 20E18, Secondary 11M41, 20C15, 20C33, 20D06, 20F69 |
| url | https://arxiv.org/abs/2604.21720 |