Generating the liftable outer automorphism groups of regular cyclic covers
Abstract
Let Rn = bigveei=1n S1 be a finite connected one-dimensional CW complex with n ≥ 2, and let Out(Fn) be the outer automorphism group of the free group Fn, where π1(Rn)≅ Fn. Let pk:R̃n→ Rn be the standard k-sheeted regular cyclic cover of Rn induced by a ℤk-action on R̃n (which is homotopy equivalent to Rnk, where nk=k(n−1)+1) generated by a free rotation through 2π/k of R̃n. Let OutFnk(Fn) be the liftable outer automorphism group associated with a standard k-sheeted normal cover pk. For n≥3, we show that {OutFnk(Fn)}k≥2 forms an infinite family of self-normalizing subgroups of Out(Fn). For n≥2, these groups are maximal if and only if k is prime. Furthermore, we derive explicit finite generating sets for OutFnk(Fn) for k≥ 2 and n≥ 3. Let 𝒞 be the infinite chain of roses, and let qn:𝒞⟶ Rn be the standard infinite-sheeted cover induced by ⟨ τ⟩, where τ is the unit translation on 𝒞. As a final application, we obtain a finite generating set for OutF∞(Fn) when n≥ 3.
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BibTeX
@misc{mahure2026generating,
title = {{Generating the liftable outer automorphism groups of regular cyclic covers}},
author = {Rohan Suresh Mahure and Priyavrat Pandey and Kashyap Rajeevsarathy},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01444},
eprint = {2611.01444},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01444},
}
RIS
TY - UNPB
TI - Generating the liftable outer automorphism groups of regular cyclic covers
AU - Rohan Suresh Mahure
AU - Priyavrat Pandey
AU - Kashyap Rajeevsarathy
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01444
UR - https://arxiv.org/abs/2611.01444
AB - Let R_n = bigvee_i=1^n S^1 be a finite connected one-dimensional CW complex with n ≥ 2, and let Out(F_n) be the outer automorphism group of the free group F_n, where π_1(R_n)≅ F_n. Let p_k:R̃_n→ R_n be the standard k-sheeted regular cyclic cover of R_n induced by a ℤ_k-action on R̃_n (which is homotopy equivalent to R_n_k, where n_k=k(n−1)+1) generated by a free rotation through 2π/k of R̃_n. Let Out_F_n_k(F_n) be the liftable outer automorphism group associated with a standard k-sheeted normal cover p_k. For n≥3, we show that {Out_F_n_k(F_n)}_k≥2 forms an infinite family of self-normalizing subgroups of Out(F_n). For n≥2, these groups are maximal if and only if k is prime. Furthermore, we derive explicit finite generating sets for Out_F_n_k(F_n) for k≥ 2 and n≥ 3. Let 𝒞 be the infinite chain of roses, and let q_n:𝒞⟶ R_n be the standard infinite-sheeted cover induced by ⟨ τ⟩, where τ is the unit translation on 𝒞. As a final application, we obtain a finite generating set for Out_F_∞(F_n) when n≥ 3.
ER -