Skip to main content

Fundamental Domains for the Maximal Groups

  • 977 Accesses

Part of the Lecture Notes in Mathematics book series (LNM,volume 2113)

Abstract

The classification of split three-dimensional crystallographic groups from Theorem 4.2 shows that seven of the groups contain all of the others as subgroups. For i = 1, , 7, we let \(\varGamma _{i} =\langle L_{i},H_{i}\rangle\), where L i is the ith lattice (in the order that the, where L i is the ith lattice (in the order that the lattices are listed in Table 4.1) and H i is the maximal point group to be paired with L i . For instance,

$$\displaystyle{\varGamma _{4} = \left \langle \left \langle \frac{1} {2}(\mathbf{x} + \mathbf{z}),\mathbf{y},\mathbf{z}\right \rangle,D_{2}^{+} \times (-1)\right \rangle.}$$

Keywords

  • Dihedral Angle
  • Unit Normal Vector
  • Fundamental Domain
  • Stabilizer Group
  • Convex Polyhedron

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (Canada)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   49.99
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions
Fig. 6.1
Fig. 6.2
Fig. 6.3
Fig. 6.4
Fig. 6.5
Fig. 6.6
Fig. 6.7

References

  1. A. Hatcher, Algebraic Topology (Cambridge University Press, Cambridge, 2002)

    Google Scholar 

  2. J. Ratcliffe, Foundations of Hyperbolic Manifolds. Graduate Texts in Mathematics, vol. 149 (Springer, New York, 1994)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and Permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Farley, D.S., Ortiz, I.J. (2014). Fundamental Domains for the Maximal Groups. In: Algebraic K-theory of Crystallographic Groups. Lecture Notes in Mathematics, vol 2113. Springer, Cham. https://doi.org/10.1007/978-3-319-08153-3_6

Download citation