The isometry group of phylogenetic tree space is 𝑆_{𝑛}

Grindstaff, Gillian

doi:10.1090/proc/15154

The isometry group of phylogenetic tree space is $S_n$
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by Gillian Grindstaff PDF

Proc. Amer. Math. Soc. 148 (2020), 4225-4233 Request permission

Abstract:

A phylogenetic tree is an acyclic graph with distinctly labeled leaves whose internal edges have a positive weight. Given a set $\{1,2,\dots ,n\}$ of $n$ leaves, the collection of all phylogenetic trees with this leaf set can be assembled into a metric cube complex known as phylogenetic tree space, or Billera-Holmes-Vogtmann tree space. In this largely combinatorial paper, we show that the isometry group of this space is the symmetric group $S_n$. This fact is relevant to the analysis of some statistical tests of phylogenetic trees, such as those introduced in Testing to distinguish measures on metric spaces, preprint, arXiv:1802.01152, 2018.

References

Alex Abreu and Marco Pacini, The automorphism group of $M_{0,n}^\textrm {trop}$ and $\overline M_{0,n}^\textrm {trop}$, J. Combin. Theory Ser. A 154 (2018), 583–597. MR 3718078, DOI 10.1016/j.jcta.2017.10.001
Dennis Barden and Huiling Le, The logarithm map, its limits and Fréchet means in orthant spaces, Proc. Lond. Math. Soc. (3) 117 (2018), no. 4, 751–789. MR 3873134, DOI 10.1112/plms.12149
Louis J. Billera, Susan P. Holmes, and Karen Vogtmann, Geometry of the space of phylogenetic trees, Adv. in Appl. Math. 27 (2001), no. 4, 733–767. MR 1867931, DOI 10.1006/aama.2001.0759
Andrew J. Blumberg, Prithwish Bhaumik, and Stephen G. Walker, Testing to distinguish measures on metric spaces, preprint, arXiv:1802.01152, 2018.
Debra L. Boutin, Identifying graph automorphisms using determining sets, Electron. J. Combin. 13 (2006), no. 1, Research Paper 78, 12. MR 2255420
Corey Bregman, Isometry groups of CAT(0) cube complexes, arXiv:1712.04805, 2017.
Dmitri Burago, Yuri Burago, and Sergei Ivanov, A course in metric geometry, Graduate Studies in Mathematics, vol. 33, American Mathematical Society, Providence, RI, 2001. MR 1835418, DOI 10.1090/gsm/033
José Cáceres, Delia Garijo, Antonio González, Alberto Márquez, and María Luz Puertas, The determining number of Kneser graphs, Discrete Math. Theor. Comput. Sci. 15 (2013), no. 1, 1–14. MR 3022925
P. Erdős, Chao Ko, and R. Rado, Intersection theorems for systems of finite sets, Quart. J. Math. Oxford Ser. (2) 12 (1961), 313–320. MR 140419, DOI 10.1093/qmath/12.1.313
Chris Godsil and Gordon Royle, Algebraic graph theory, Graduate Texts in Mathematics, vol. 207, Springer-Verlag, New York, 2001. MR 1829620, DOI 10.1007/978-1-4613-0163-9
Susan Holmes, Statistical approach to tests involving phylogenies, Mathematics of Evolution and Phylogeny (2005), 91–120.
Bo Lin, Anthea Monod, and Ruriko Yoshida, Tropical foundations for probability & statistics on phylogenetic tree space, preprint, arXiv:1805.12400, 2018.
Michah Sageev, $\rm CAT(0)$ cube complexes and groups, Geometric group theory, IAS/Park City Math. Ser., vol. 21, Amer. Math. Soc., Providence, RI, 2014, pp. 7–54. MR 3329724, DOI 10.1090/pcms/021/02
Amy Willis, Confidence sets for phylogenetic trees, J. Amer. Statist. Assoc. 114 (2019), no. 525, 235–244. MR 3941251, DOI 10.1080/01621459.2017.1395342