Let $PB_n$ be the $n$-strand pure braid group. For each $1\le k\le n$, let $\kappa_k^n \colon PB_n \to PB_{n+1}$ be the monomorphism that takes a pure braid and replaces the $k$th strand with two parallel strands. (So, "strand bifurcation".) Suppose we have a subgroup $H_n \le PB_n$ for each $n$. Call the sequence $(H_n)_{n\in\mathbb{N}}$ coherent if the image $\kappa_k^n(H_n)$ lies in $H_{n+1}$ for all $1\le k\le n$. Call the sequence complete if the preimage $(\kappa_k^n)^{-1}(H_{n+1})$ lies in $H_n$ for all $1\le k\le n$.

Question: What are some examples of coherent, complete sequences $(H_n)_{n\in\mathbb{N}}$? I specifically want examples where each $H_n$ is normal, not only in $PB_n$ but even in the braid group $B_n$.

The only examples I've ever managed to think of (in https://arxiv.org/abs/1403.8132) are as follows. For $m\in\mathbb{N}$, call a pure braid $m$-loose if it becomes trivial upon deleting all but any $m$ strands. (If $m>n$ this doesn't really make sense, but just declare that the identity in $PB_n$ is $m$-loose and nothing else is.) For example, all pure braids are 1-loose, the 2-loose pure braids in $PB_n$ comprise the commutator subgroup $PB_n'$, and so forth. If $L_n^{(m)}$ is the subgroup of all $m$-loose elements of $PB_n$, then it turns out $\big(L_n^{(m)}\big)_{n\in\mathbb{N}}$ is an example of such a sequence. (Note that each $m$ provides such a sequence.)

The motivation for all this is, it turns out the normal subgroups of the Brin-Dehornoy braided Thompson group $bV$ are precisely classified by such sequences. It is a longstanding problem to figure out whether $bV$ is Hopfian, and so simply finding more examples of such sequences is a good first step. (If it somehow happened that the above $m$-loose sequences were the only examples, then I can see an approach to proving $bV$ is Hopfian, but this feels like too much to hope for.)

$\DeclareMathOperator\PB{PB}\DeclareMathOperator\B{B}$Let $\PB_n$ be the $n$-strand pure braid group. For each $1\le k\le n$, let $\kappa_k^n \colon \PB_n \to \PB_{n+1}$ be the monomorphism that takes a pure braid and replaces the $k$th strand with two parallel strands. (So, "strand bifurcation".) Suppose we have a subgroup $H_n \le \PB_n$ for each $n$. Call the sequence $(H_n)_{n\in\mathbb{N}}$ coherent if the image $\kappa_k^n(H_n)$ lies in $H_{n+1}$ for all $1\le k\le n$. Call the sequence complete if the preimage $(\kappa_k^n)^{-1}(H_{n+1})$ lies in $H_n$ for all $1\le k\le n$.

Question: What are some examples of coherent, complete sequences $(H_n)_{n\in\mathbb{N}}$? I specifically want examples where each $H_n$ is normal, not only in $\PB_n$ but even in the braid group $\B_n$.

The only examples I've ever managed to think of (in https://arxiv.org/abs/1403.8132) are as follows. For $m\in\mathbb{N}$, call a pure braid $m$-loose if it becomes trivial upon deleting all but any $m$ strands. (If $m>n$ this doesn't really make sense, but just declare that the identity in $\PB_n$ is $m$-loose and nothing else is.) For example, all pure braids are 1-loose, the 2-loose pure braids in $\PB_n$ comprise the commutator subgroup $\PB_n'$, and so forth. If $L_n^{(m)}$ is the subgroup of all $m$-loose elements of $\PB_n$, then it turns out $\big(L_n^{(m)}\big)_{n\in\mathbb{N}}$ is an example of such a sequence. (Note that each $m$ provides such a sequence.)

The motivation for all this is, it turns out the normal subgroups of the Brin-Dehornoy braided Thompson group $bV$ are precisely classified by such sequences. It is a longstanding problem to figure out whether $bV$ is Hopfian, and so simply finding more examples of such sequences is a good first step. (If it somehow happened that the above $m$-loose sequences were the only examples, then I can see an approach to proving $bV$ is Hopfian, but this feels like too much to hope for.)