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Peer-reviewedCC BYIsrael Journal of Mathematics

Probabilistic laws on infinite groups

  • Gideon Amir1
  • Guy Blachar1
  • Maria Gerasimova2
  • Gady Kozma3
  1. 1Department of Mathematics, Bar-Ilan University, Ramat Gan, 52900, Israel
  2. 2Institut für Informatik, Westälische Wilhelms-Universität Münster, 48149, Münster, Germany
  3. 3Department of Mathematics, The Weizmann Institute of Science, Rehovot, 76100, Israel
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Abstract

We study the probability that certain laws are satisfied on infinite groups, focusing on elements sampled by random walks. For several group laws, including the metabelian one, we construct examples of infinite groups for which the law holds with high probability, but the group does not satisfy the law virtually. On the other hand, we show that if an infinite group satisfies the law x 2 = 1 with positive probability, then it is virtually abelian.

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Cite

BibTeXRIS
BibTeX
@article{amir2026probabilistic,
  title = {{Probabilistic laws on infinite groups}},
  author = {Gideon Amir and Guy Blachar and Maria Gerasimova and Gady Kozma},
  journal = {Israel Journal of Mathematics},
  year = {2026},
  month = oct,
  doi = {10.1007/s11856-026-2943-2},
  url = {https://doi.org/10.1007/s11856-026-2943-2},
  note = {CC BY},
}
RIS
TY  - JOUR
TI  - Probabilistic laws on infinite groups
AU  - Gideon Amir
AU  - Guy Blachar
AU  - Maria Gerasimova
AU  - Gady Kozma
PY  - 2026
DA  - 2026/10/19
JO  - Israel Journal of Mathematics
DO  - 10.1007/s11856-026-2943-2
UR  - https://doi.org/10.1007/s11856-026-2943-2
AB  - We study the probability that certain laws are satisfied on infinite groups, focusing on elements sampled by random walks. For several group laws, including the metabelian one, we construct examples of infinite groups for which the law holds with high probability, but the group does not satisfy the law virtually. On the other hand, we show that if an infinite group satisfies the law x 2 = 1 with positive probability, then it is virtually abelian.
SN  - 0021-2172
SN  - 1565-8511
ER  -