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Generic Rigidity with Forced Symmetry and Sparse Colored Graphs

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Book cover Rigidity and Symmetry

Part of the book series: Fields Institute Communications ((FIC,volume 70))

Abstract

We review some recent results in the generic rigidity theory of planar frameworks with forced symmetry, giving a uniform treatment to the topic. We also give new combinatorial characterizations of minimally rigid periodic frameworks with fixed-area fundamental domain and fixed-angle fundamental domain.

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Notes

  1. 1.

    See, e.g., the recent conferences [9, 21, 40].

  2. 2.

    Infinite frameworks with no other assumptions can exhibit quite complicated behavior [33].

  3. 3.

    We are extending the terminology “Teichmüller space” from its more typical usage for the group \(\mathbb{Z}^{2}\) and lattices in \(\mathrm{PSL}(2, \mathbb{R})\). Our definition of \(\text{Teich}(\mathbb{Z}^{2})\) is non-standard since the usual one allows only unit-area fundamental domains.

  4. 4.

    Colored graphs are also known as “gain graphs” or “voltage graphs” [53]. The terminology of colored graphs originates from [36].

  5. 5.

    The sparsity counts we describe here are slightly different from what is stated in [38, Theorem 4.2.1], but they are equivalent by an argument similar to that in the proof of Proposition 4. This presentation highlights the connection to colored-Laman graphs.

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Acknowledgements

We thank the Fields Institute for its hospitality during the Workshop on Rigidity and Symmetry, the workshop organizers for putting together the program, and the conference participants for many interesting discussions. LT is supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no 247029-SDModels. JM partially supported by NSF CDI-I grant DMR 0835586 and (for finial preparation) the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no 226135.

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Authors and Affiliations

  1. Einstein Institute of Mathematics, Edmond J. Safra Campus, Givat Ram, The Hebrew University of Jerusalem, Jerusalem, Israel

    Justin Malestein

  2. Institut für Mathematik, Freie Universitüt Berlin, Berlin, Germany

    Louis Theran

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  1. Justin Malestein
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Correspondence to Justin Malestein .

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  1. Department of Mathematics, Cornell University, Ithaca, New York, USA

    Robert Connelly

  2. Department of Mathematics and Statistics, York University, Toronto, Ontario, Canada

    Asia Ivić Weiss

  3. Department of Mathematics & Statistics, York University, Toronto, Ontario, Canada

    Walter Whiteley

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Malestein, J., Theran, L. (2014). Generic Rigidity with Forced Symmetry and Sparse Colored Graphs. In: Connelly, R., Ivić Weiss, A., Whiteley, W. (eds) Rigidity and Symmetry. Fields Institute Communications, vol 70. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0781-6_12

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