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PreprintCC BY-SA 4.0arXiv

Images of word maps with constants on algebraic groups

  • Vadim Alekseev
  • Jakob Schneider
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Abstract

We study word maps with constants on quasisimple algebraic groups over a local field L. We prove that, for such a group G=G(L) and a word w∈ (G∗Fr)∖ G, either w has a so-called Tomanov-small critical constant or the minimal dimension of the word image w(Gr)⊆ G of such a word is bounded from below by a function c(G)>1. We compute the optimal value of c(G) for most quasisimple linear algebraic groups.

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Cite

BibTeXRIS
BibTeX
@misc{alekseev2026images,
  title = {{Images of word maps with constants on algebraic groups}},
  author = {Vadim Alekseev and Jakob Schneider},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.09058},
  eprint = {2609.09058},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.09058},
  note = {CC BY-SA 4.0},
}
RIS
TY  - UNPB
TI  - Images of word maps with constants on algebraic groups
AU  - Vadim Alekseev
AU  - Jakob Schneider
PY  - 2026
DA  - 2026/09/08
PB  - arXiv
DO  - 10.48550/arxiv.2609.09058
UR  - https://arxiv.org/abs/2609.09058
AB  - We study word maps with constants on quasisimple algebraic groups over a local field L. We prove that, for such a group G=G(L) and a word w∈ (G∗F_r)∖ G, either w has a so-called Tomanov-small critical constant or the minimal dimension of the word image w(G^r)⊆ G of such a word is bounded from below by a function c(G)>1. We compute the optimal value of c(G) for most quasisimple linear algebraic groups.
ER  -