Chip-Firing Games and Critical Groups

Part of the Foundations for Undergraduate Research in Mathematics book series (FURM)


In this note we introduce a finite abelian group that can be associated with any finite connected graph. This group can be defined in an elementary combinatorial way in terms of chip-firing operations, and has been an object of interest in combinatorics, algebraic geometry, statistical physics, and several other areas of mathematics. We will begin with basic definitions and examples and develop a number of properties that can be derived by looking at this group from different angles. Throughout, we will give exercises, some of which are straightforward and some of which are open questions. We will also highlight some of the many contributions to this area made by undergraduate students.



We would like to thank Luis David Garcia-Puente for initiating this project. We would further like to thank David Jensen, Pranav Kayastha, Dino Lorenzini, Sam Payne, Farbod Shokrieh, and the editors and referees for their helpful comments.

The second author is supported by NSF Grant DMS 1802281.


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© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Gettysburg CollegeGettysburgUSA
  2. 2.University of CaliforniaIrvineUSA

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