arXiv subjects

Jeremy Macdonald

Publications and source records attributed to Jeremy Macdonald.

5 recordsLinked to original sources

arXiv2021

Logspace and compressed-word computations in nilpotent groups

For finitely generated nilpotent groups, we employ Mal'cev coordinates to solve several classical algorithmic problems efficiently. Computation of normal forms, the membership problem, the conjugacy problem, and computation of presentations for subgroups are solved using only logarithmic space and quasilinear time. Logarithmic space presentation-uniform versions of these algorithms are provided. Compressed-word versions of the same problems, in which each input word is provided as a straight-line program, are solved in polynomial time.

math.GR↗

arXiv2017

Low-complexity computations for nilpotent subgroup problems

We solve the following algorithmic problems using TC0 circuits, or in logspace and quasilinear time, uniformly in the class of nilpotent groups with bounded nilpotency class and rank: subgroup conjugacy, computing the normalizer and isolator of a subgroup, coset intersection, and computing the torsion subgroup. Additionally, if any input words are provided in compressed form as straight-line programs or in Mal'cev coordinates the algorithms run in quartic time.

math.GR↗

arXiv2014

Effective coherence of groups discriminated by a locally quasi-convex hyperbolic group

We prove that every finitely generated group $G$ discriminated by a locally quasi-convex torsion-free hyperbolic group $Γ$ is effectively coherent: that is, presentations for finitely generated subgroups can be computed from the subgroup generators. We study $G$ via its embedding into an iterated centralizer extension of $Γ$, and prove that this embedding can be computed. We also give algorithms to enumerate all finitely generated groups discriminated by $Γ$ and to decide whether a given group, with decidable word problem, is discriminated by $Γ$. If $Γ$ may have torsion, we prove that groups obtained from $Γ$ by iterated amalgamated products with virtually abelian groups, over elementary subgroups, are effectively coherent.

math.GR↗

arXiv2013

Effective embedding of residually hyperbolic groups into direct products of extensions of centralizers

For any torsion-free hyperbolic group $Γ$ and any group $G$ that is fully residually $Γ$, we construct algorithmically a finite collection of homomorphisms from $G$ to groups obtained from $Γ$ by extensions of centralizers, at least one of which is injective. When $G$ is residually $Γ$, this gives a effective embedding of $G$ into a direct product of such groups. We also give an algorithmic construction of a diagram encoding the set of homomorphisms from a given finitely presented group to $Γ$.

math.GR↗

arXiv2009

Compressed words and automorphisms in fully residually free groups

We show that the compressed word problem in a finitely-generated fully residually free group (F -group) is decidable in polynomial time, and use the result to show that the word problem in the automorphism group of such a group is decidable in polynomial time.

math.GR↗