For g≥2, let Mod(Sg) be the mapping class group of the closed orientable surface Sg of genus g. In this paper, we provide necessary and sufficient conditions for a pair of elements in Mod(Sg) to generate an infinite metacyclic subgroup. In particular, we provide necessary and sufficient conditions under which a pseudo-Anosov mapping class generates an infinite metacyclic subgroup of Mod(Sg) with a nontrivial periodic mapping class. As applications of our main results, we establish the existence of infinite metacyclic subgroups of Mod(Sg) isomorphic to Z⋊Zm, Zn⋊Z, and Z⋊Z. Furthermore, we derive bounds on the order of a nontrivial periodic generator of an infinite metacyclic subgroup of Mod(Sg) that are realized. Finally, we show that the centralizer of an irreducible periodic mapping class F is either 〈F〉 or 〈F〉×〈i〉, where i is a hyperelliptic involution.
1
Indian Institute of Science Education and Research Bhopal, Madhya Pradesh, India
Pankaj Kapari; Kashyap Rajeevsarathy; Apeksha Sanghi. Infinite metacyclic subgroups of the mapping class group. Groups, geometry, and dynamics, Tome 19 (2025) no. 1, pp. 281-313. doi: 10.4171/ggd/791
@article{10_4171_ggd_791,
author = {Pankaj Kapari and Kashyap Rajeevsarathy and Apeksha Sanghi},
title = {Infinite metacyclic subgroups of the mapping class group},
journal = {Groups, geometry, and dynamics},
pages = {281--313},
year = {2025},
volume = {19},
number = {1},
doi = {10.4171/ggd/791},
url = {http://geodesic.mathdoc.fr/articles/10.4171/ggd/791/}
}
TY - JOUR
AU - Pankaj Kapari
AU - Kashyap Rajeevsarathy
AU - Apeksha Sanghi
TI - Infinite metacyclic subgroups of the mapping class group
JO - Groups, geometry, and dynamics
PY - 2025
SP - 281
EP - 313
VL - 19
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.4171/ggd/791/
DO - 10.4171/ggd/791
ID - 10_4171_ggd_791
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%T Infinite metacyclic subgroups of the mapping class group
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%D 2025
%P 281-313
%V 19
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%R 10.4171/ggd/791
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