# How long does it take for the action of the braid monoids on Laver tables to become trivial?

Let $A_{n}$ denote the classical Laver table of cardinality $2^{n}$.

Let $B_{n}^{*}$ denote the positive (including the identity) braid monoid on $n$ elements generated by $\sigma_{1},...,\sigma_{n}$. Let $BW_{n}^{*}$ denote the collection of all words over the alphabet $\sigma_{1},...,\sigma_{n}$ including the empty word.

Then whenever $(X,*)$ is a self-distributive structure, then the monoid $B_{n}$ acts on $X^{n+1}$ by letting $\sigma_{i}(x_{1},...,x_{i},x_{i+1},x_{n+1})=(x_{1},...,x_{i}*x_{i+1},x_{i},...,x_{n+1})$ (in other words, we replace $(x_{i},x_{i+1})$ with $(x_{i}*x_{i+1},x_{i})$).

Suppose now that $n,m$ are natural numbers. Then let $T(n,m)$ denote the largest natural number such that there is a positive braid word $\sigma$ such that $|\sigma|\geq T(n,m)$ along with $x_{1},...,x_{n+1}\in A_{m}$ such that if $\sigma(x_{1},...,x_{n+1})=(y_{1},...,y_{n+1})$, then $y_{i}<2^{m}$ whenever $1\leq i\leq n+1$. Here $|\sigma|$ denotes the length of the braid word $\sigma$.

What are some good bounds for the size of $T(n,m)$? The function $T(n,m)$ is increasing in both variables. Furthermore, by a fairly strightforward argument, I have obtained the lower bounds $T(n,n)\geq n(n-1)$.

If for all $n$ there exists an $n$-huge cardinal, one can prove that for all $n$, we have $^{\lim}_{m\rightarrow\infty}T(n,m)\rightarrow\infty$ (this limit however seems to grow very slowly). However, is there a ZFC proof of this result? For a given $n$, how slowly does the function $m\mapsto T(n,m)$ grow if it does converge to $\infty$?

Let $L(m,n)$ be the largest natural number such that there exists $x_{1},...,x_{n+1}\in A_{m}$ where $|\{\sigma(x_{1},...,x_{n+1})|\sigma\in BW_{n}^{+}\}|=L(m,n)$. I am also interested in lower and upper bounds for $L(m,n)$.

The motivation behind this question is that I want to see how applicable the Laver tables would be to the study of braids and knots, and a good way to test the applicability of Laver tables to braids and knots is to see how non-trivial the action of positive braids on the Laver tables is. This question is a part of a long term project to make large cardinals more applicable to diverse areas of mathematics through the study of algebras of elementary embeddings.