The contraction category of graphs

Proudfoot, Nicholas; Ramos, Eric

doi:10.1090/ert/616

The contraction category of graphs
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by Nicholas Proudfoot and Eric Ramos

Represent. Theory 26 (2022), 673-697

DOI: https://doi.org/10.1090/ert/616

Published electronically: June 28, 2022

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Abstract:

We study the category whose objects are graphs of fixed genus and whose morphisms are contractions. We show that the corresponding contravariant module categories are Noetherian and we study two families of modules over these categories. The first takes a graph to a graded piece of the homology of its unordered configuration space and the second takes a graph to an intersection homology group whose dimension is given by a Kazhdan–Lusztig coefficient; in both cases we prove that the module is finitely generated. This allows us to draw conclusions about torsion in the homology groups of graph configuration spaces, and about the growth of Betti numbers of graph configuration spaces and Kazhdan–Lusztig coefficients of graphical matroids. We also explore the relationship between our category and outer space, which is used in the study of outer automorphisms of free groups.

References

Aaron David Abrams, Configuration spaces and braid groups of graphs, ProQuest LLC, Ann Arbor, MI, 2000. Thesis (Ph.D.)–University of California, Berkeley. MR 2701024
Byung Hee An, Gabriel C. Drummond-Cole, and Ben Knudsen, Subdivisional spaces and graph braid groups, Doc. Math. 24 (2019), 1513–1583. MR 4033833
Daniel Barter, Noetherianity and rooted trees, Preprint, arXiv:1509.04228, 2015
Thomas Church, Jordan S. Ellenberg, and Benson Farb, FI-modules and stability for representations of symmetric groups, Duke Math. J. 164 (2015), no. 9, 1833–1910. MR 3357185, DOI 10.1215/00127094-3120274
Marc Culler and Karen Vogtmann, Moduli of graphs and automorphisms of free groups, Invent. Math. 84 (1986), no. 1, 91–119. MR 830040, DOI 10.1007/BF01388734
Jan Draisma, Noetherianity up to symmetry, Combinatorial algebraic geometry, Lecture Notes in Math., vol. 2108, Springer, Cham, 2014, pp. 33–61. MR 3329086, DOI 10.1007/978-3-319-04870-3_{2}
Ben Elias, Nicholas Proudfoot, and Max Wakefield, The Kazhdan-Lusztig polynomial of a matroid, Adv. Math. 299 (2016), 36–70. MR 3519463, DOI 10.1016/j.aim.2016.05.005
Michael Farber, Invitation to topological robotics, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2008. MR 2455573, DOI 10.4171/054
Ki Hyoung Ko and Hyo Won Park, Characteristics of graph braid groups, Discrete Comput. Geom. 48 (2012), no. 4, 915–963. MR 3000570, DOI 10.1007/s00454-012-9459-8
Daniel Lütgehetmann, Representation stability for configuration spaces of graphs, Preprint, arXiv:1701.03490, 2019.
C. St. J. A. Nash-Williams, On well-quasi-ordering finite trees, Proc. Cambridge Philos. Soc. 59 (1963), 833–835. MR 153601, DOI 10.1017/s0305004100003844
Peter Orlik and Louis Solomon, Combinatorics and topology of complements of hyperplanes, Invent. Math. 56 (1980), no. 2, 167–189. MR 558866, DOI 10.1007/BF01392549
Nicholas Proudfoot, The algebraic geometry of Kazhdan-Lusztig-Stanley polynomials, EMS Surv. Math. Sci. 5 (2018), no. 1-2, 99–127. MR 3880222, DOI 10.4171/EMSS/28
Nicholas Proudfoot and Eric Ramos, Functorial invariants of trees and their cones, Selecta Math. (N.S.) 25 (2019), no. 4, Paper No. 62, 28. MR 4021848, DOI 10.1007/s00029-019-0509-4
Nicholas Proudfoot, Max Wakefield, and Ben Young, Intersection cohomology of the symmetric reciprocal plane, J. Algebraic Combin. 43 (2016), no. 1, 129–138. MR 3439303, DOI 10.1007/s10801-015-0628-8
Nicholas Proudfoot and Ben Young, Configuration spaces, $\rm FS^{op}$-modules, and Kazhdan-Lusztig polynomials of braid matroids, New York J. Math. 23 (2017), 813–832. MR 3690232
Andrew Putman, Stability in the homology of congruence subgroups, Invent. Math. 202 (2015), no. 3, 987–1027. MR 3425385, DOI 10.1007/s00222-015-0581-0
Eric Ramos, An application of the theory of FI-algebras to graph configuration spaces, Math. Z. 294 (2020), no. 1-2, 1–15. MR 4050061, DOI 10.1007/s00209-019-02278-w
Eric Ramos and Graham White, Families of nested graphs with compatible symmetric-group actions, Selecta Math. (N.S.) 25 (2019), no. 5, Paper No. 70, 42. MR 4030224, DOI 10.1007/s00029-019-0520-9
Steven V. Sam and Andrew Snowden, Gröbner methods for representations of combinatorial categories, J. Amer. Math. Soc. 30 (2017), no. 1, 159–203. MR 3556290, DOI 10.1090/jams/859
John Smillie and Karen Vogtmann, A generating function for the Euler characteristic of $\textrm {Out}(F_n)$, Proceedings of the Northwestern conference on cohomology of groups (Evanston, Ill., 1985), 1987, pp. 329–348. MR 885116, DOI 10.1016/0022-4049(87)90036-3
Andrew Snowden, Syzygies of Segre embeddings and $\Delta$-modules, Duke Math. J. 162 (2013), no. 2, 225–277. MR 3018955, DOI 10.1215/00127094-1962767
Karen Vogtmann, Automorphisms of free groups and outer space, Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part I (Haifa, 2000), 2002, pp. 1–31. MR 1950871, DOI 10.1023/A:1020973910646
Karen Vogtmann, The cohomology of automorphism groups of free groups, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 1101–1117. MR 2275637
Peter Webb, An introduction to the representations and cohomology of categories, Group representation theory, EPFL Press, Lausanne, 2007, pp. 149–173. MR 2336640
Bruno Zimmermann, Finite groups of outer automorphisms of free groups, Glasgow Math. J. 38 (1996), no. 3, 275–282. MR 1417356, DOI 10.1017/S0017089500031700