On the automorphism group of the canonical double covering of bordered Klein surfaces with large automorphism group

AbstractWe discuss the classifiation of simply connected, complete (κ, µ)-spaces from the point of view of homogeneous spaces. In particular, we exhibit new models of (κ, µ)-spaces having Boeckx invariant -1. Finally, we prove that the number ${{(n + 1)(n + 2)} \over 2}$ is the maximum dimension of the automorphism group of a contact metric manifold of dimension 2n +1, n ≥ 2, whose symmetric operator h has rank at least 3 at some point; if this dimension is attained, and the dimension of the manifold is not 7, it must be a (κ, µ)-space. The same conclusion holds also in dimension 7 provided the almost CR structure of the contact metric manifold under consideration is integrable.

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Complex doubles of bordered Klein surfaces with maximal symmetry

Glasgow Mathematical Journal ◽

10.1017/s0017089500008041 ◽

1991 ◽

Vol 33 (1) ◽

pp. 61-71 ◽

Cited By ~ 6

Author(s):

Coy L. May

Keyword(s):

Automorphism Group ◽

Special Interest ◽

Klein Surface ◽

Klein Surfaces ◽

Maximal Symmetry

A compact bordered Klein surface X of algebraic genus g ≥ 2 has maximal symmetry [6] if its automorphism group A(X) is of order 12(g — 1), the largest possible. The bordered surfaces with maximal symmetry are clearly of special interest and have been studied in several recent papers ([6] and [9] among others).

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4-dimensional Minkowski planes with large automorphism group

Forum Mathematicum ◽

10.1515/form.1992.4.593 ◽

1992 ◽

Vol 4 (4) ◽

Cited By ~ 3

Author(s):

Günter F. Steinke

Keyword(s):

Automorphism Group ◽

Minkowski Planes ◽

Large Automorphism Group

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Symmetries of Riemann surfaces with large automorphism group

Mathematische Annalen ◽

10.1007/bf01344543 ◽

1974 ◽

Vol 210 (1) ◽

pp. 17-32 ◽

Cited By ~ 50

Author(s):

David Singerman

Keyword(s):

Automorphism Group ◽

Riemann Surfaces ◽

Large Automorphism Group

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On the automorphism group of hyperelliptic Klein surfaces.

10.1307/mmj/1029003817 ◽

1988 ◽

Vol 35 (3) ◽

pp. 361-368 ◽

Cited By ~ 1

Author(s):

E. Bujalance ◽

J. A. Bujalance ◽

E. Martínez

Keyword(s):

Automorphism Group ◽

Klein Surfaces

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THE SYMMETRIC CROSSCAP NUMBER OF THE GROUPS OF SMALL-ORDER

Journal of Algebra and Its Applications ◽

10.1142/s0219498812501642 ◽

2012 ◽

Vol 12 (02) ◽

pp. 1250164 ◽

Cited By ~ 4

Author(s):

J. J. ETAYO ◽

E. MARTÍNEZ

Keyword(s):

Finite Group ◽

Automorphism Group ◽

The Other ◽

Klein Surfaces ◽

Exceptional Groups ◽

Minimal Genus ◽

Other Hand ◽

Unique Integer ◽

Symmetric Crosscap Number

Every finite group G acts as an automorphism group of some non-orientable Klein surfaces without boundary. The minimal genus of these surfaces is called the symmetric crosscap number and denoted by [Formula: see text]. It is known that 3 cannot be the symmetric crosscap number of a group. Conversely, it is also known that all integers that do not belong to nine classes modulo 144 are the symmetric crosscap number of some group. Here we obtain infinitely many groups whose symmetric crosscap number belong to each one of six of these classes. This result supports the conjecture that 3 is the unique integer which is not the symmetric crosscap number of a group. On the other hand, there are infinitely many groups with symmetric crosscap number 1 or 2. For g > 2 the number of groups G with [Formula: see text] is finite. The value of [Formula: see text] is known when G belongs to certain families of groups. In particular, if o(G) < 32, [Formula: see text] is known for all except thirteen groups. In this work we obtain it for these groups by means of a one-by-one analysis. Finally we obtain the least genus greater than two for those exceptional groups whose symmetric crosscap number is 1 or 2.

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