scholarly journals An Efficient Algorithm for Finding a Basis of the Fixed Point Subgroup of an Automorphism of a Free Group

2014 ◽  
pp. 17-23
Author(s):  
Oleg Bogopolski ◽  
Olga Maslakova
Keyword(s):  
Fixed Point ◽  
Free Group ◽  
Efficient Algorithm

Rings with Fixed-Point-Free Group Actions

1973 ◽  
Vol s3-27 (1) ◽  
pp. 69-87 ◽  
Author(s):  
G. M. Bergman ◽  
I. M. Isaacs
Keyword(s):  
Fixed Point ◽  
Free Group ◽  
Group Actions ◽  
Free Group Actions

Centraliser near-rings determined by fixed point free automorphism groups

1987 ◽  
Vol 107 (3-4) ◽  
pp. 327-337 ◽  
Author(s):  
Peter Fuchs ◽  
C. J. Maxson ◽  
M. R. Pettet ◽  
K. C. Smith
Keyword(s):  
Fixed Point ◽  
Free Group ◽  
Automorphism Groups ◽  
Group Of Automorphisms ◽  
Nontrivial Ideal

SynopsisLet G be a group and let A be a fixed point free group of automorphisms of G. It is shown that the centraliser near-ring MA(G) has at most one nontrivial ideal. Conditions on the pair (A, G) are given which force MA(G) to be simple. It is shown that if a nonsimple near-ring MA(G) exists, then A and G have unusual properties.

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

Cybernetics ◽  
1982 ◽  
Vol 17 (3) ◽  
pp. 407-416 ◽  
Author(s):  
A. A. Letichevskii ◽  
A. B. Godlevskii ◽  
S. L. Krivoi
Keyword(s):  
Free Group ◽  
Efficient Algorithm

An algorithm for finding a basis of the fixed point subgroup of an automorphism of a free group

2016 ◽  
Vol 26 (01) ◽  
pp. 29-67 ◽  
Author(s):  
Oleg Bogopolski ◽  
Olga Maslakova
Keyword(s):  
Fixed Point ◽  
Free Group ◽  
Finite Rank

We describe an algorithm which, given an automorphism [Formula: see text] of a free group [Formula: see text] of finite rank, computes a basis of the fixed point subgroup [Formula: see text].

scholarly journals THE CANCELLATION NORM AND THE GEOMETRY OF BI-INVARIANT WORD METRICS

2015 ◽  
Vol 58 (1) ◽  
pp. 153-176 ◽  
Author(s):  
MICHAEL BRANDENBURSKY ◽  
ŚWIATOSŁAW R. GAL ◽  
JAREK KĘDRA ◽  
MICHAŁ MARCINKOWSKI
Keyword(s):  
Cayley Graph ◽  
Free Group ◽  
Efficient Algorithm ◽  
Isometric Embedding ◽  
Finitely Generated ◽  
Locally Compact ◽  
Cyclic Subgroups ◽  
Geometric Origin ◽  
Word Norm

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.

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

2007 ◽  
Vol 17 (02) ◽  
pp. 289-328 ◽  
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.

scholarly journals Tight Convergence Rate of Gradient Descent for Eigenvalue Computation

2020 ◽  
Author(s):  
Qinghua Ding ◽  
Kaiwen Zhou ◽  
James Cheng
Keyword(s):  
Fixed Point ◽  
Convergence Rate ◽  
Efficient Algorithm ◽  
Gradient Descent ◽  
Convergence Rates ◽  
Real Data ◽  
Convergence Results ◽  
Fast Convergence Rate ◽  
Quadratic Factor ◽  
Prior Analysis

Riemannian gradient descent (RGD) is a simple, popular and efficient algorithm for leading eigenvector computation [AMS08]. However, the existing analysis of RGD for eigenproblem is still not tight, which is O(log(n/epsilon)/Delta^2) due to [Xu et al., 2018]. In this paper, we show that RGD in fact converges at rate O(log(n/epsilon)/Delta), and give instances to shows the tightness of our result. This improves the best prior analysis by a quadratic factor. Besides, we also give tight convergence analysis of a deterministic variant of Oja's rule due to [Oja, 1982]. We show that it also enjoys fast convergence rate of O(log(n/epsilon)/Delta). Previous papers only gave asymptotic characterizations [Oja, 1982; Oja, 1989; Yi et al., 2005]. Our tools for proving convergence results include an innovative reduction and chaining technique, and a noisy fixed point iteration argument. Besides, we also give empirical justifications of our convergence rates over synthetic and real data.

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