## 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

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## Bibliographic Information

**Nicholas Proudfoot**- Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403
- MR Author ID: 689525
- Email: njp@uoregon.edu
**Eric Ramos**- Affiliation: Department of Mathematics, Bowdoin College, Brunswick, Maine 04011
- MR Author ID: 993563
- Email: e.ramos@bowdoin.edu
- Received by editor(s): July 18, 2020
- Received by editor(s) in revised form: January 18, 2022, and April 21, 2022
- Published electronically: June 28, 2022
- Additional Notes: This work was supported by NSF grants DMS-1704811, DMS-1954050, DMS-2039316, DMS-2053243, and DMS-2137628.
- © Copyright 2022 American Mathematical Society
- Journal: Represent. Theory
**26**(2022), 673-697 - MSC (2020): Primary 05C25, 05C83, 14F43, 55R80
- DOI: https://doi.org/10.1090/ert/616
- MathSciNet review: 4445717