Peer-reviewedCC BY-NCInventiones Mathematicae
Growth of permutational extensions
Abstract
We study the geometry of a class of group extensions, containing permutational wreath products, which we call “permutational extensions”. We construct for all k∈ℕ a torsion group K k with growth function vKk(n)∼exp(n1−(1−α)k), 23−3/α+22−2/α+21−1/α=2, and a torsion-free group H k with growth function vHk(n)∼exp(log(n)n1−(1−α)k). These are the first examples of groups of intermediate growth for which the asymptotics of their growth function is known. We construct a group of intermediate growth that contains the group of finitely supported permutations of a countable set as a subgroup. This gives the first example of a group of intermediate growth containing an infinite simple group as a subgroup.
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BibTeX
@article{bartholdi2026growth,
title = {{Growth of permutational extensions}},
author = {Laurent Bartholdi and Anna Erschler},
journal = {Inventiones Mathematicae},
year = {2026},
month = nov,
doi = {10.1007/s00222-011-0368-x},
eprint = {1011.5266},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05849588v1},
note = {CC BY-NC},
}
RIS
TY - JOUR
TI - Growth of permutational extensions
AU - Laurent Bartholdi
AU - Anna Erschler
PY - 2026
DA - 2026/11/03
JO - Inventiones Mathematicae
DO - 10.1007/s00222-011-0368-x
UR - https://hal.science/hal-05849588v1
AB - We study the geometry of a class of group extensions, containing permutational wreath products, which we call “permutational extensions”. We construct for all k∈ℕ a torsion group K k with growth function v_K_k(n)∼exp(n^1−(1−α)^k), 2^3−3/α+2^2−2/α+2^1−1/α=2, and a torsion-free group H k with growth function v_H_k(n)∼exp(log(n)n^1−(1−α)^k). These are the first examples of groups of intermediate growth for which the asymptotics of their growth function is known. We construct a group of intermediate growth that contains the group of finitely supported permutations of a countable set as a subgroup. This gives the first example of a group of intermediate growth containing an infinite simple group as a subgroup.
SN - 0020-9910
SN - 1432-1297
ER -