Non-solvable torsion-free virtually solvable groups

Jonathan A. Hillman

Non-solvable torsion-free virtually solvable groups

Group Theory 2025-06-05 v8

Authors: Jonathan A. Hillman

Abstract

There are perfect Bieberbach groups of Hirsch length 15, but none in lower dimensions. We shall show that a nonsolvable, torsion free, virtually solvable group $S$ must have Hirsch length $h(S)\geq10$ . If $h(S)\leq13$ then we may assume that $A_5$ is the only simple factor, but $PSL(2,7)$ and $SL(2,8)$ may occur when $h(S)\geq14$ . There are no known examples with $h(S)<15$ .

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@article{arxiv.2302.09513,
  title  = {Non-solvable torsion-free virtually solvable groups},
  author = {Jonathan A. Hillman},
  journal= {arXiv preprint arXiv:2302.09513},
  year   = {2025}
}

Comments

v2: Results in \S5 and \S7 sharpened. Reference [5] updated. v3: New \S8, using torsion in crystallographic quotients. v4:Theorem 20 corrected, \S8 expanded. v5: Lemmas 4 and 7, and Sections 5 and 8 rewritten. v6: Reorganised. v: PSL(2,7) and SL(2,8) excluded in dimensions $<14$

Non-solvable torsion-free virtually solvable groups

Abstract

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