scholarly journals Numerical studies of Thompson’s group F and related groups

2019 ◽  
Vol 29 (02) ◽  
pp. 179-243
Author(s):  
Andrew Elvey Price ◽  
Anthony J. Guttmann
Keyword(s):  
Lower Bounds ◽  
Numerical Data ◽  
Universality Class ◽  
Finite Graph ◽  
Computational Approach ◽  
Locally Finite Graph ◽  
Lamplighter Group ◽  
Thompson’S Group ◽  
Thompson Group ◽  
Thompson's Group F

We have developed polynomial-time algorithms to generate terms of the cogrowth series for groups [Formula: see text], the lamplighter group, [Formula: see text] and the Brin–Navas group [Formula: see text]. We have also given an improved algorithm for the coefficients of Thompson’s group [Formula: see text], giving 32 terms of the cogrowth series. We develop numerical techniques to extract the asymptotics of these various cogrowth series. We present improved rigorous lower bounds on the growth-rate of the cogrowth series for Thompson’s group F using the method from [S. Haagerup, U. Haagerup and M. Ramirez-Solano, A computational approach to the Thompson group F, Int. J. Alg. Comp. 25 (2015) 381–432] applied to our extended series. We also generalise their method by showing that it applies to loops on any locally finite graph. Unfortunately, lower bounds less than 16 do not help in determining amenability. Again for Thompson’s group F we prove that, if the group is amenable, there cannot be a sub-dominant stretched exponential term in the asymptotics. Yet the numerical data provides compelling evidence for the presence of such a term. This observation suggests a potential path to a proof of non-amenability: If the universality class of the cogrowth sequence can be determined rigorously, it will likely prove non-amenability. We estimate the asymptotics of the cogrowth coefficients of F to be [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text]. The growth constant [Formula: see text] must be 16 for amenability. These two approaches, plus a third based on extrapolating lower bounds, support the conjecture [M. Elder, A. Rechnitzer and E. J. Janse van Rensburg, Random sampling of trivial words in finitely presented groups, Expr. Math. 24 (2015) 391–409, S. Haagerup, U. Haagerup and M. Ramirez-Solano, A computational approach to the Thompson group F, Int. J. Alg. Comp. 25 (2015) 381–432] that the group is not amenable.

Thompson’s Group F

2018 ◽  
pp. 37-66
Author(s):  
Marianna C. Bonanome ◽  
Margaret H. Dean ◽  
Judith Putnam Dean
Keyword(s):  
Thompson’S Group ◽  
Thompson's Group F

scholarly journals Supersymmetry and multicriticality in a ladder of constrained fermions

2021 ◽  
Vol 11 (3) ◽  
Author(s):  
Natalia Chepiga ◽  
Jiří Minář ◽  
Kareljan Schoutens
Keyword(s):  
Phase Diagram ◽  
Density Wave ◽  
Valence Bond ◽  
Lattice Models ◽  
Numerical Data ◽  
Universality Class ◽  
Density Profiles ◽  
Valence Bond Solid ◽  
Floating Phase ◽  
Solid Type

Supersymmetric lattice models of constrained fermions are known to feature exotic phenomena such as superfrustration, with an extensive degeneracy of ground states, the nature of which is however generally unknown. Here we address this issue by considering a superfrustrated model, which we deform from the supersymetric point. By numerically studying its two-parameter phase diagram, we reveal a rich phenomenology. The vicinity of the supersymmetric point features period-4 and period-5 density waves which are connected by a floating phase (incommensurate Luttinger liquid) with smoothly varying density. The supersymmetric point emerges as a multicritical point between these three phases. Inside the period-4 phase we report a valence-bond solid type ground state that persists up to the supersymmetric point. Our numerical data for transitions out of density-wave phases are consistent with the Pokrovsky-Talapov universality class. Furthermore, our analysis unveiled a period-3 phase with a boundary determined by a competition between single and two-particle instabilities accompanied by a doubling of the wavevector of the density profiles along a line in the phase diagram.

scholarly journals Lower Bounds for Identifying Codes in some Infinite Grids

10.37236/394 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Ryan Martin ◽  
Brendon Stanton
Keyword(s):  
Lower Bounds ◽  
Finite Graph ◽  
Infinite Graphs ◽  
Locally Finite ◽  
Identifying Codes ◽  
Identifying Code ◽  
Hexagonal Grids ◽  
Closed Neighborhood ◽  
Definition Of ◽  
Infinite Grids

An $r$-identifying code on a graph $G$ is a set $C\subset V(G)$ such that for every vertex in $V(G)$, the intersection of the radius-$r$ closed neighborhood with $C$ is nonempty and unique. On a finite graph, the density of a code is $|C|/|V(G)|$, which naturally extends to a definition of density in certain infinite graphs which are locally finite. We present new lower bounds for densities of codes for some small values of $r$ in both the square and hexagonal grids.

scholarly journals On the density of Cayley graphs of R.Thompson’s group F in symmetric generators

2021 ◽  
pp. 1-13
Author(s):  
V. S. Guba
Keyword(s):  
Cayley Graph ◽  
Cayley Graphs ◽  
Finite Graph ◽  
Vertex Degree ◽  
Doubling Property ◽  
Generating Set ◽  
Finite Set ◽  
Standard Set ◽  
Thompson’S Group ◽  
Average Vertex

By the density of a finite graph we mean its average vertex degree. For an [Formula: see text]-generated group, the density of its Cayley graph in a given set of generators, is the supremum of densities taken over all its finite subgraphs. It is known that a group with [Formula: see text] generators is amenable if and only if the density of the corresponding Cayley graph equals [Formula: see text]. A famous problem on the amenability of R. Thompson’s group [Formula: see text] is still open. Due to the result of Belk and Brown, it is known that the density of its Cayley graph in the standard set of group generators [Formula: see text], is at least [Formula: see text]. This estimate has not been exceeded so far. For the set of symmetric generators [Formula: see text], where [Formula: see text], the same example only gave an estimate of [Formula: see text]. There was a conjecture that for this generating set equality holds. If so, [Formula: see text] would be non-amenable, and the symmetric generating set would have the doubling property. This would mean that for any finite set [Formula: see text], the inequality [Formula: see text] holds. In this paper, we disprove this conjecture showing that the density of the Cayley graph of [Formula: see text] in symmetric generators [Formula: see text] strictly exceeds [Formula: see text]. Moreover, we show that even larger generating set [Formula: see text] does not have doubling property.

Asymptotic Behavior of Skew Conditional Heat Kernels on Graph Networks

1993 ◽  
Vol 45 (4) ◽  
pp. 863-878 ◽  
Author(s):  
Tatsuya Okada
Keyword(s):  
Asymptotic Behavior ◽  
Finite Graph ◽  
Heat Propagation ◽  
Heat Kernels ◽  
Locally Finite ◽  
Locally Finite Graph ◽  
Periodic Models ◽  
Definition Of

AbstractIn this note, we will consider the heat propagation on locally finite graph networks which satisfy a skew condition on vertices (See Definition of Section 2). For several periodic models, we will construct the heat kernels Pt with the skew condition explicitly, and derive the decay order of Pt as time goes to infinity.

scholarly journals ON THE PROPERTIES OF THE CAYLEY GRAPH OF RICHARD THOMPSON'S GROUP F

2004 ◽  
Vol 14 (05n06) ◽  
pp. 677-702 ◽  
Author(s):  
V. S. GUBA
Keyword(s):  
Growth Rate ◽  
Lower Bound ◽  
Cayley Graph ◽  
Upper Bound ◽  
Growth Function ◽  
Vertex Degree ◽  
General Growth ◽  
Thompson’S Group ◽  
Average Vertex ◽  
Thompson's Group F

We study some properties of the Cayley graph of R. Thompson's group F in generators x0, x1. We show that the density of this graph, that is, the least upper bound of the average vertex degree of its finite subgraphs is at least 3. It is known that a 2-generated group is not amenable if and only if the density of the corresponding Cayley graph is strictly less than 4. It is well known this is also equivalent to the existence of a doubling function on the Cayley graph. This means there exists a mapping from the set of vertices into itself such that for some constant K>0, each vertex moves by a distance at most K and each vertex has at least two preimages. We show that the density of the Cayley graph of a 2-generated group does not exceed 3 if and only if the group satisfies the above condition with K=1. Besides, we give a very easy formula to find the length (norm) of a given element of F in generators x0, x1. This simplifies the algorithm by Fordham. The length formula may be useful for finding the general growth function of F in generators x0, x1 and the growth rate of this function. In this paper, we show that the growth rate of F has a lower bound of [Formula: see text].

scholarly journals Random subgroups of Thompson’s group F

2010 ◽  
pp. 91-126 ◽  
Author(s):  
Sean Cleary ◽  
Murray Elder ◽  
Andrew Rechnitzer ◽  
Jennifer Taback
Keyword(s):  
Thompson’S Group ◽  
Thompson's Group F

scholarly journals On Jones’ subgroup of R. Thompson’s group T

2018 ◽  
Vol 28 (05) ◽  
pp. 877-903
Author(s):  
Jordan Nikkel ◽  
Yunxiang Ren
Keyword(s):  
Unitary Representation ◽  
Unitary Representations ◽  
Thompson Groups ◽  
Diagram Groups ◽  
Thompson’S Group ◽  
Planar Algebra ◽  
Thompson Group

Jones introduced unitary representations for the Thompson groups [Formula: see text] and [Formula: see text] from a given subfactor planar algebra. Some interesting subgroups arise as the stabilizer of certain vector, in particular the Jones subgroups [Formula: see text] and [Formula: see text]. Golan and Sapir studied [Formula: see text] and identified it as a copy of the Thompson group [Formula: see text]. In this paper, we completely describe [Formula: see text] and show that [Formula: see text] coincides with its commensurator in [Formula: see text], implying that the corresponding unitary representation is irreducible. We also generalize the notion of the Stallings 2-core for diagram groups to [Formula: see text], showing that [Formula: see text] and [Formula: see text] are not isomorphic, but as annular diagram groups they have very similar presentations.

Sign in / Sign up

Export Citation Format

Share Document