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## On the cogrowth of Thompson’s group

Venue: | F . Groups Complex. Cryptol |

Citations: | 5 - 3 self |

### Citations

2223 |
Iterative methods for sparse linear systems
- Saad
- 2003
(Show Context)
Citation Context ...excluding (1,−) have outdegree at most 3 (vertices on the boundary may have smaller degree) the corresponding adjacency matrices are sparse. We found that the power method and Rayleigh quotients (see =-=[20]-=- for example) converged very quickly to the dominant eigenvalue and so the growth rate. We constructed GN and HN for many different values of N ranging between 102 and 107. Our calculations on Thompso... |

517 |
The On-Line Encyclopedia of Integer Sequences. Published electronically at http://www.research.att.com/sequences
- Sloane
- 2009
(Show Context)
Citation Context ...byproduct of our computations we obtained the first few terms of the cogrowth series for all of these groups. It is well know that the number of trivial words in Z2 is given by ( 2n n )2 (see A002894 =-=[22]-=-); the corresponding generating function is not algebraic and is expressible as a complete elliptic integral of the first kind. The number of trivial words in F2 is just the number of returning paths ... |

164 | Gfun: a Maple package for the manipulation of generating and holonomic functions in one variable
- Salvy, Zimmermann
- 1994
(Show Context)
Citation Context ...s in F2 is just the number of returning paths in a quadtree and its generating function is 3(1 + 2 √ 1− 12z2)−1 (see A035610 [22]). Unfortunately we have been unable to find (using tools such as GFUN =-=[21]-=-) any useful explicit or implicit expressions for the cogrowth series (or the generating functions) for any of the other groups we have examined. For completeness we include our data in Table 3. 3 Dis... |

43 |
Non-amenable finitely presented torsion-by-cyclic groups, Publications Mathmatiques de l’IHES 96
- Ol’shanskii, Sapir
- 2003
(Show Context)
Citation Context ...cted a finitely generated non-amenable group with no nonabelian free subgroups [14], and in 1982 Adyan gave further examples [1]. In 2002 Ol’shanskii and Sapir constructed finitely presented examples =-=[15]-=-. In spite of these results the amenability or non-amenability of F remains an intensely studied problem. In the second half of the article we extend our techniques to study the distribution of geodes... |

40 |
Cogrowth and amenability of discrete groups
- Cohen
- 1982
(Show Context)
Citation Context ...r generators and their inverses to label distinct edges in G, then ρ ≤ 2k. The connection between this growth rate and amenability was established by Grigorchuk and independently by Cohen: Theorem 1 (=-=[12, 7]-=-). Let G,S and ρ be as above. G is amenable if and only if ρ = 2k. Let pn be the number of returns of length n on G which do not contain immediate reversals. Again concatenation shows that pn is super... |

38 |
Minimal length elements of Thompson’s group
- Fordham
(Show Context)
Citation Context ... Thompson’s group and a number of different wreath products. • Thompson’s group — a method for computing the geodesic length of an element from its tree-pair representation was first given by Fordham =-=[4]-=-, though we found it easier to implement the method of Belk and Brown [3]. • Wreath products — we use the results of [6] to find the geodesic lengths in Z o Z, Z o (Z o Z) and Z o F2. We note that the... |

28 |
On the question of the existence of an invariant mean on a group
- Ol’shanskii
- 1980
(Show Context)
Citation Context ...onjecture that a group is non-amenable if and only if it contains a nonabelian free subgroup. In 1980 Ol’shanskii constructed a finitely generated non-amenable group with no nonabelian free subgroups =-=[14]-=-, and in 1982 Adyan gave further examples [1]. In 2002 Ol’shanskii and Sapir constructed finitely presented examples [15]. In spite of these results the amenability or non-amenability of F remains an ... |

27 | Forest diagrams for elements of Thompson’s group
- Belk, Brown
(Show Context)
Citation Context ... group — a method for computing the geodesic length of an element from its tree-pair representation was first given by Fordham [4], though we found it easier to implement the method of Belk and Brown =-=[3]-=-. • Wreath products — we use the results of [6] to find the geodesic lengths in Z o Z, Z o (Z o Z) and Z o F2. We note that the geodesic problem for Baumslag-Solitar groups has recently been solved in... |

25 | Growth series of some wreath products
- Parry
- 1992
(Show Context)
Citation Context ...vial rate of escape. The group Z3 o Z2 is amenable but has positive rate of escape [19]. Unfortunately, computing geodesics in this group is equivalent to solving the traveling salesman problem on Z3 =-=[16]-=- and so beyond these techniques. 15 Figure 4: A plot of the normalised distribution of the number of words cn,` of length n and geodesic length ` in Z o Z. Observe that the peak position is clearly mo... |

23 |
Symmetrical random walks on discrete groups
- Grigorchuk
- 1980
(Show Context)
Citation Context ...r generators and their inverses to label distinct edges in G, then ρ ≤ 2k. The connection between this growth rate and amenability was established by Grigorchuk and independently by Cohen: Theorem 1 (=-=[12, 7]-=-). Let G,S and ρ be as above. G is amenable if and only if ρ = 2k. Let pn be the number of returns of length n on G which do not contain immediate reversals. Again concatenation shows that pn is super... |

22 |
Random walks on free periodic groups
- Adyan
- 1982
(Show Context)
Citation Context ...only if it contains a nonabelian free subgroup. In 1980 Ol’shanskii constructed a finitely generated non-amenable group with no nonabelian free subgroups [14], and in 1982 Adyan gave further examples =-=[1]-=-. In 2002 Ol’shanskii and Sapir constructed finitely presented examples [15]. In spite of these results the amenability or non-amenability of F remains an intensely studied problem. In the second half... |

22 |
Rate of escape of random walks on wreath products and related
- Revelle
(Show Context)
Citation Context ...te of escape implies that Thompson’s group is non-amenable, however there are examples of amenable groups with nontrivial rate of escape. The group Z3 o Z2 is amenable but has positive rate of escape =-=[19]-=-. Unfortunately, computing geodesics in this group is equivalent to solving the traveling salesman problem on Z3 [16] and so beyond these techniques. 15 Figure 4: A plot of the normalised distribution... |

10 |
2004 Flat Histogram Version of the Pruned and Enriched
- Prellberg, Krawczyk
(Show Context)
Citation Context ...simple sampling will decay exponentially quickly. We will proceed along a similar line but using a more powerful random sampling method based on flat-histogram ideas used in the FlatPERM algorithm 12 =-=[17, 18]-=-. Each sample word is grown in a similar manner to simple sampling — append one generator at a time chosen uniformly at random. The weight of a word of n symbols is simply 1, so that the total weight ... |

9 |
Analytic combinatorics. Cambridge Univ Pr
- Flajolet, Sedgewick
- 2009
(Show Context)
Citation Context ...,n exists by Fekete’s lemma. Further we must have rn ≥ rN,n and so ρ ≥ ρN . Hence we can bound ρ by computing ρN . Using the Perron-Frobenius theorem (in one of its many guises — Proposition V.7 from =-=[11]-=- for example) the growth rate ρN of such paths on GN is given by the dominant eigenvalue of the corresponding adjacency matrix, provided it is irreducible. We construct GN so that it is connected and ... |

9 |
A course in combinatorics, Cambridge Univ Pr
- Lint, Wilson
- 2001
(Show Context)
Citation Context ...g and ending at the identity element — let us call such paths returns. Since we can concatenate any two such paths to get another we have rnrk ≤ rn+k (1) and then by Fekete’s lemma (see, for example, =-=[23]-=-) ρ = lim sup n→∞ r 1/n n (2) ∗The first author acknowledges support from ARC projects DP110101104 and FT110100178. The second author thanks NSERC of Canada for financial support. 1Formally, we consid... |

8 | A linear-time algorithm to compute geodesics in solvable Baumslag-solitar groups
- Elder
(Show Context)
Citation Context ...— we use the results of [6] to find the geodesic lengths in Z o Z, Z o (Z o Z) and Z o F2. We note that the geodesic problem for Baumslag-Solitar groups has recently been solved in the cases BS(1, n) =-=[10]-=- and BS(n, kn) [8], but we have not implemented these approaches. 3.1 Distributions We used the random sampling algorithm described above to estimate the distribution of geodesic lengths in Thompson’s... |

7 | Metric properties of the lamplighter group as an automata group, Geometric Methods
- Cleary, Taback
- 2005
(Show Context)
Citation Context ...ngth of an element from its tree-pair representation was first given by Fordham [4], though we found it easier to implement the method of Belk and Brown [3]. • Wreath products — we use the results of =-=[6]-=- to find the geodesic lengths in Z o Z, Z o (Z o Z) and Z o F2. We note that the geodesic problem for Baumslag-Solitar groups has recently been solved in the cases BS(1, n) [10] and BS(n, kn) [8], but... |

7 | On rationality of the cogrowth series
- Kouksov
- 1998
(Show Context)
Citation Context ...ly reduced path since it may create an immediate reversal. Thus we do not have similar supermultiplicative relations. We can, however, relate rn to pn and ρ to α using the following result of Kouksov =-=[13]-=- which we have specialised to the case of 2 generator groups. 4 Lemma 3. (from [13]) Let R(z) = ∑ rnz n and C(z) = ∑ pnz n be the generating functions of returns and freely reduced returns respectivel... |

5 | On computing geodesics in Baumslag-Solitar groups
- Diekert, Laun
(Show Context)
Citation Context ...s of [6] to find the geodesic lengths in Z o Z, Z o (Z o Z) and Z o F2. We note that the geodesic problem for Baumslag-Solitar groups has recently been solved in the cases BS(1, n) [10] and BS(n, kn) =-=[8]-=-, but we have not implemented these approaches. 3.1 Distributions We used the random sampling algorithm described above to estimate the distribution of geodesic lengths in Thompson’s group F , as well... |

4 | Computational explorations in Thompson’s group
- Burillo, Cleary, et al.
- 2007
(Show Context)
Citation Context ...he second half of the article we extend our techniques to study the distribution of geodesic words in Thompson’s group. This work is in the same spirit as previous papers by Burillo, Cleary and Wiest =-=[5]-=-, and Arzhantseva, Guba, Lustig, and Préaux [2], who also applied 2 computational techniques to consider the amenability of F . We refer the reader to these papers for more background on Thompson’s g... |

4 |
Lower bounds for the spectral radii of adjacency operators on Baumslag-Solitar groups, Arxiv preprint arXiv:1006.0556
- Dykema, Redelmeier
- 2010
(Show Context)
Citation Context ...y 5. We observed that the bounds obtained from GN were worse — typically differing in the second or third significant digit. We also note that the above result for BS(2, 3) is improves on a result in =-=[9]-=- (the preprint was withdrawn by the authors since it contained an error). These computations were done on a desktop computer using about 4Gb of memory. It should be noted that while our techniques req... |

3 | Polymer Simulations with a Flat Histogram Stochastic Growth Algorithm, in: Computer simulation studies in condensed-matter physics XVI: proceedings of the seventeenth workshop
- Prellberg, Krawczyk, et al.
(Show Context)
Citation Context ...simple sampling will decay exponentially quickly. We will proceed along a similar line but using a more powerful random sampling method based on flat-histogram ideas used in the FlatPERM algorithm 12 =-=[17, 18]-=-. Each sample word is grown in a similar manner to simple sampling — append one generator at a time chosen uniformly at random. The weight of a word of n symbols is simply 1, so that the total weight ... |

1 |
Testing Cayley graph densities. (Tester les densités de graphes de Cayley
- Arzhantseva, Guba, et al.
(Show Context)
Citation Context ...niques to study the distribution of geodesic words in Thompson’s group. This work is in the same spirit as previous papers by Burillo, Cleary and Wiest [5], and Arzhantseva, Guba, Lustig, and Préaux =-=[2]-=-, who also applied 2 computational techniques to consider the amenability of F . We refer the reader to these papers for more background on Thompson’s group and the problem of deciding its amenability... |