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Semigroup Intersection Problems in the Heisenberg Groups

Author Ruiwen Dong

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Ruiwen Dong
  • Department of Computer Science, University of Oxford, UK

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Ruiwen Dong. Semigroup Intersection Problems in the Heisenberg Groups. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 25:1-25:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


We consider two algorithmic problems concerning sub-semigroups of Heisenberg groups and, more generally, two-step nilpotent groups. The first problem is Intersection Emptiness, which asks whether a finite number of given finitely generated semigroups have empty intersection. This problem was first studied by Markov in the 1940s. We show that Intersection Emptiness is PTIME decidable in the Heisenberg groups H_n(𝕂) over any algebraic number field 𝕂, as well as in direct products of Heisenberg groups. We also extend our decidability result to arbitrary finitely generated 2-step nilpotent groups. The second problem is Orbit Intersection, which asks whether the orbits of two matrices under multiplication by two semigroups intersect with each other. This problem was first studied by Babai et al. (1996), who showed its decidability within commutative matrix groups. We show that Orbit Intersection is decidable within the Heisenberg group H₃(ℚ).

Subject Classification

ACM Subject Classification
  • Computing methodologies → Symbolic and algebraic manipulation
  • semigroup intersection
  • orbit intersection
  • matrix semigroups
  • Heisenberg group
  • nilpotent groups


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