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Arithmetic Groups, Base Change, and Representation Growth
| Content Provider | Semantic Scholar |
|---|---|
| Author | Avni, Nir Klopsch, Benjamin Voll, Christopher |
| Copyright Year | 2011 |
| Abstract | AbstractConsider an arithmetic group $${\mathbf{G}(O_S)}$$G(OS), where $${\mathbf{G}}$$G is an affine group scheme with connected, simply connected absolutely almost simple generic fiber, defined over the ring of S-integers OS of a number field K with respect to a finite set of places S. For each $${n \in \mathbb{N}}$$n∈N, let $${R_n(\mathbf{G}(O_S))}$$Rn(G(OS)) denote the number of irreducible complex representations of $${\mathbf{G}(O_S)}$$G(OS) of dimension at most n. The degree of representation growth $${\alpha(\mathbf{G}(O_S)) = \lim_{n \rightarrow\infty} \log R_n(\mathbf{G}(O_S)) / \log n}$$α(G(OS))=limn→∞logRn(G(OS))/logn is finite if and only if $${\mathbf{G}(O_S)}$$G(OS) has the weak Congruence Subgroup Property. We establish that for every $${\mathbf{G}(O_S)}$$G(OS) with the weak Congruence Subgroup Property the invariant $${\alpha(\mathbf{G}(O_S))}$$α(G(OS)) is already determined by the absolute root system of $${\mathbf{G}}$$G. To show this we demonstrate that the abscissae of convergence of the representation zeta functions of such groups are invariant under base extensions $${K{\subset}L}$$K⊂L. We deduce from our result a variant of a conjecture of Larsen and Lubotzky regarding the representation growth of irreducible lattices in higher rank semi-simple groups. In particular, this reduces Larsen and Lubotzky’s conjecture to Serre’s conjecture on the weak Congruence Subgroup Property, which it refines. |
| Starting Page | 67 |
| Ending Page | 135 |
| Page Count | 69 |
| File Format | PDF HTM / HTML |
| DOI | 10.1007/s00039-016-0359-6 |
| Volume Number | 26 |
| Alternate Webpage(s) | https://arxiv.org/pdf/1110.6092v4.pdf |
| Alternate Webpage(s) | https://doi.org/10.1007/s00039-016-0359-6 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
