An efficient algorithm for constructing the basis of a subgroup of a free group

AbstractWe study bi-invariant word metrics on groups. We provide an efficient algorithm for computing the bi-invariant word norm on a finitely generated free group and we construct an isometric embedding of a locally compact tree into the bi-invariant Cayley graph of a nonabelian free group. We investigate the geometry of cyclic subgroups. We observe that in many classes of groups, cyclic subgroups are either bounded or detected by homogeneous quasimorphisms. We call this property the bq-dichotomy and we prove it for many classes of groups of geometric origin.

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POLYNOMIAL-TIME COMPLEXITY FOR INSTANCES OF THE ENDOMORPHISM PROBLEM IN FREE GROUPS

International Journal of Algebra and Computation ◽

10.1142/s0218196707003597 ◽

2007 ◽

Vol 17 (02) ◽

pp. 289-328 ◽

Cited By ~ 3

Author(s):

LAURA CIOBANU

Keyword(s):

Free Group ◽

Polynomial Time ◽

Efficient Algorithm ◽

Time Complexity ◽

Free Groups ◽

Polynomial Time Algorithm ◽

Time Algorithm ◽

The Other ◽

Polynomial Time Complexity

We say the endomorphism problem is solvable for an element W in a free group F if it can be decided effectively whether, given U in F, there is an endomorphism ϕ of F sending W to U. This work analyzes an approach due to Edmunds and improved by Sims. Here we prove that the approach provides an efficient algorithm for solving the endomorphism problem when W is a two-generator word. We show that when W is a two-generator word this algorithm solves the problem in time polynomial in the length of U. This result gives a polynomial-time algorithm for solving, in free groups, two-variable equations in which all the variables occur on one side of the equality and all the constants on the other side.

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Bases and Automorphisms of the Free Group on Two Generators

From Christoffel Words to Markoff Numbers ◽

10.1093/oso/9780198827542.003.0017 ◽

2018 ◽

pp. 125-142

Author(s):

Christophe Reutenauer

Keyword(s):

Free Group ◽

Efficient Algorithm ◽

Group Of Automorphisms ◽

Inner Automorphism

The chapter begins with a self-contained exposition of the theory of Nielsen on the free groupwith two generators: bases of F(a, b),Nielsen’s criterion for automorphisms of F(a, b), It also coversNielsen’s theoremon abelianization of these automorphisms andWeinbaum’s theorem on representatives of the group of automorphisms modulo the subgroup of inner automorphism. Perrine’s theorem on bases of the derived group of SL2(Z) and Markoff triples is deduced, and a very simple and efficient algorithm for detecting bases of F(a, b) is given (Séébold, Kassel, the author). Positive automorphisms of F(a, b) are characterized (Wen andWen) and shown to coincide with Sturmian morphisms (Mignosi, Séébold).

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RANDOM GENERATION OF FINITELY GENERATED SUBGROUPS OF A FREE GROUP

International Journal of Algebra and Computation ◽

10.1142/s0218196708004482 ◽

2008 ◽

Vol 18 (02) ◽

pp. 375-405 ◽

Cited By ~ 4

Author(s):

FRÉDÉRIQUE BASSINO ◽

CYRIL NICAUD ◽

PASCAL WEIL

Keyword(s):

Uniform Distribution ◽

Free Group ◽

Finite Index ◽

Finite Rank ◽

Efficient Algorithm ◽

Random Generation ◽

Finitely Generated ◽

Average Case ◽

Case Complexity ◽

Average Case Complexity

We give an efficient algorithm to randomly generate finitely generated subgroups of a given size, in a finite rank free group. Here, the size of a subgroup is the number of vertices of its representation by a reduced graph such as can be obtained by the method of Stallings foldings. Our algorithm randomly generates a subgroup of a given size n, according to the uniform distribution over size n subgroups. In the process, we give estimates of the number of size n subgroups, of the average rank of size n subgroups, and of the proportion of such subgroups that have finite index. Our algorithm has average case complexity [Formula: see text] in the RAM model and [Formula: see text] in the bitcost model.

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Computational Micrograph Registration with Sieve Processes

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100164660 ◽

1996 ◽

Vol 54 ◽

pp. 440-441

Author(s):

P.J. Phillips ◽

J. Huang ◽

S. M. Dunn

Keyword(s):

Planar Graph ◽

Efficient Algorithm ◽

Spatial Organization ◽

Spatial Arrangement ◽

Stereo Image ◽

Image Model ◽

Geometric Invariants ◽

Point Set

In this paper we present an efficient algorithm for automatically finding the correspondence between pairs of stereo micrographs, the key step in forming a stereo image. The computation burden in this problem is solving for the optimal mapping and transformation between the two micrographs. In this paper, we present a sieve algorithm for efficiently estimating the transformation and correspondence.In a sieve algorithm, a sequence of stages gradually reduce the number of transformations and correspondences that need to be examined, i.e., the analogy of sieving through the set of mappings with gradually finer meshes until the answer is found. The set of sieves is derived from an image model, here a planar graph that encodes the spatial organization of the features. In the sieve algorithm, the graph represents the spatial arrangement of objects in the image. The algorithm for finding the correspondence restricts its attention to the graph, with the correspondence being found by a combination of graph matchings, point set matching and geometric invariants.

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