Nilpotent automorphism groups of bordered Klein surfaces

Author:
Coy L. May

Journal:
Proc. Amer. Math. Soc. **101** (1987), 287-292

MSC:
Primary 30F35; Secondary 14H99, 20H10

MathSciNet review:
902543

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a compact bordered Klein surface of algebraic genus , and let be a group of automorphisms of . Then the order of is at most . Here we improve this general bound in an important special case. We show that if is nilpotent, then the order of is at most . This bound is the best possible. We construct infinite families of surfaces that have a nilpotent automorphism group of order . The nilpotent groups of maximum possible order must be -groups. We prove that if the nilpotent group acts on a bordered surface of genus such that , then is a power of 2. Further, our examples show that for each nonnegative integer there is a bordered surface with genus and a group of automorphisms of order .

**[1]**E. Bujalance and J. M. Gamboa,*Automorphism groups of algebraic curves of 𝑅ⁿ of genus 2*, Arch. Math. (Basel)**42**(1984), no. 3, 229–237. MR**751500**, 10.1007/BF01191180**[2]**Newcomb Greenleaf and Coy L. May,*Bordered Klein surfaces with maximal symmetry*, Trans. Amer. Math. Soc.**274**(1982), no. 1, 265–283. MR**670931**, 10.1090/S0002-9947-1982-0670931-X**[3]**Coy L. May,*Automorphisms of compact Klein surfaces with boundary*, Pacific J. Math.**59**(1975), no. 1, 199–210. MR**0399451****[4]**Coy L. May,*Large automorphism groups of compact Klein surfaces with boundary. I*, Glasgow Math. J.**18**(1977), no. 1, 1–10. MR**0425113****[5]**-,*Supersolvable**-groups*, Glasgow Math. J. (to appear).**[6]**David Singerman,*Orientable and nonorientable Klein surfaces with maximal symmetry*, Glasgow Math. J.**26**(1985), no. 1, 31–34. MR**776674**, 10.1017/S0017089500005747**[7]**David Singerman,*𝑃𝑆𝐿(2,𝑞) as an image of the extended modular group with applications to group actions on surfaces*, Proc. Edinburgh Math. Soc. (2)**30**(1987), no. 1, 143–151. Groups—St. Andrews 1985. MR**879440**, 10.1017/S001309150001806X**[8]**Reza Zomorrodian,*Nilpotent automorphism groups of Riemann surfaces*, Trans. Amer. Math. Soc.**288**(1985), no. 1, 241–255. MR**773059**, 10.1090/S0002-9947-1985-0773059-6

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
30F35,
14H99,
20H10

Retrieve articles in all journals with MSC: 30F35, 14H99, 20H10

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1987-0902543-4

Keywords:
Bordered Klein surface,
genus,
automorphism,
nilpotent group,
-group,
full covering

Article copyright:
© Copyright 1987
American Mathematical Society