Computational explorations in Thompson's group F

An exploration of normalish subgroups of R. Thompson's groups $F$ and $T$

In this short note, we show that R. Thompson's group $F$ admits a normalish amenable subgroup, and that the standard copy of $F$ in R. Thompson's group $T$ is normalish in $T$. We further conjecture that if $F$ is non-amenable, then $T$ does not admit a normalish amenable subgroup, and therefore that the reduced $C^*$ algebra of $T$ is in fact simple in that case.

Computational explorations of the Thompson group T for the amenability problem of F

It is a long standing open problem whether the Thompson group $F$ is an amenable group. In this paper we show that if $A$, $B$, $C$ denote the standard generators of Thompson group $T$ and $D:=C B A^{-1}$ then $$\sqrt2+\sqrt3\,<\,\frac1{\sqrt{12}}||(I+C+C^2)(I+D+D^2+D^3)||\,\le\, 2+\sqrt2.$$ Moreover, the upper bound is attained if the Thompson group $F$ is amenable. Here, the norm of an element in the group ring $\mathbb{C} T$ is computed in $B(\ell^2(T))$ via the regular representation of $T$. Using the "cyclic reduced" numbers $τ(((C+C^2)(D+D^2+D^3))^n)$, $n\in\mathbb{N}$, and some methods …

One-way permutations, computational asymmetry and distortion

Computational asymmetry, i.e., the discrepancy between the complexity of transformations and the complexity of their inverses, is at the core of one-way transformations. We introduce a computational asymmetry function that measures the amount of one-wayness of permutations. We also introduce the word-length asymmetry function for groups, which is an algebraic analogue of computational asymmetry. We relate boolean circuits to words in a Thompson monoid, over a fixed generating set, in such a way that circuit size is equal to word-length. Moreover, boolean circuits have a representation in terms…

A computational approach to the Thompson group $F$

Let $F$ denote the Thompson group with standard generators $A=x_0$, $B=x_1$. It is a long standing open problem whether $F$ is an amenable group. By a result of Kesten from 1959, amenability of $F$ is equivalent to $$(i)\qquad ||I+A+B||=3$$ and to $$(ii)\qquad ||A+A^{-1}+B+B^{-1}||=4,$$ where in both cases the norm of an element in the group ring $\mathbb{C} F$ is computed in $B(\ell^2(F))$ via the regular representation of $F$. By extensive numerical computations, we obtain precise lower bounds for the norms in $(i)$ and $(ii)$, as well as good estimates of the spectral distributions of $(I+A…

The Thompson-Higman monoids M_{k,i}: the J-order, the D-relation, and their complexity

The Thompson-Higman groups G_{k,i} have a natural generalization to monoids M_{k,i}, and inverse monoids Inv_{k,i}. We study some structural features of M_{k,i} and Inv_{k,i} and investigate the computational complexity of decision problems. The main interest of these monoids is their close connection with circuits and circuit complexity. The maximal subgroups of M_{k,1} are isomorphic to the groups G_{k,j} (1 \leq j \leq k-1); so we rediscover all the Thompson-Higman groups within M_{k,1}. The Green relations \leq_J and \equiv_D of M_{k,1} can be decided in deterministic polynomial time when …

小域上李型有限单群的汤普森猜想

Thompson's Conjecture for Finite Simple Groups of Lie Type over Small Fields

该研究证明了阶不超过8的域上李型有限单群存在平方为整个群的共轭类，结合此前结果完成汤普森猜想的证明，还明确构造相关类并提供计算证书。