All the subfactors here are

irreducibleinclusion of hyperfinite II$_1$ factors.

A subfactor $(N \subset M)$ is **Homogeneous Single Chain** ($HSC$) if its lattice of intermediate subfactors is a single chain: $N=P_0 \subset P_1 \subset \dots \subset P_{n-1} \subset P_n= M$, with $(P_i \subset P_{i+1}) \simeq (P_0 \subset P_1)$.

We qualify such a subfactor by the **notation** $HSC^n_{(P_0 \subset P_1)}$, because it's a subfactor whose lattice is a single chain of length $n$ and whose intermediate maximal inclusions are all isomorphic to $(P_0 \subset P_1)$.

Question: Is there a $HSC^n_{(P \subset Q)}$ subfactor for all maximal subfactor $(P \subset Q)$ and for all $n>1$ ?

**Warning**: The subfactors given by the basic construction are not irreducible.

**Group-subgroup formulation**:

**Definition** : Let $(A \subset B)$ and $(C \subset D)$ be inclusions of groups, then $(A \subset B) \sim (C \subset D)$ if $(A/A_B \subset B/A_B) \simeq (C/C_D \subset D/C_D)$, with $A_B$ the normal core of $A$ in $B$.

**Remark** : In the finite group case, if $(A \subset B) \sim (C \subset D)$ then $(R^B \subset R^A) \simeq (R^D \subset R^C)$, but the converse is false, see here ($R$ is the hyperfinite II$_1$ factor).

**Definition**: An inclusion of groups is $HSC^n_{(A \subset B)}$ if its lattice of intermediate subgroups is a single chain of length $n$ and if its intermediate maximal inclusions of groups are all $\sim$ to $(A \subset B)$.

Question: Is there a $HSC^n_{(A \subset B)}$ inclusion for all maximal inclusion $(A \subset B)$ and for all $n>1$ ?

**Remark**: An inclusion of group $(A \subset B)$ is given (up to $\sim$) by a transitive permutation group $G$ of degree $d = [A:B]$ (i.e. $G$ is the image of the coset representation and $(A \subset B) \sim (G_1 \subset G$)).

Idem a maximal inclusion of groups is given (up to $\sim$) by a primitive permutation group.

**Notation**: A transitive group $G$ is $HSC^n_{M}$ (with $M$ a primitive group) if $(G_1 \subset G)$ is $HSC^n_{(M_1 \subset M)}$.

**Experimental answer**: **Yes**, if we restrict to the transitive groups of degree $\le 30$.

More precisely, let $G_{d,r}$ be the $r$-th transitive group of degree $d$ given by the GAP Data Library :

```
gap> TransitiveGroup(5,4);
A5
```

**Remark**: $G_{d,1}$ is the cyclic group $\mathbb{Z}/d\mathbb{Z}$

The primitive groups of deg. $\le 5$: $G_{2,1}$ ; $G_{3,1}$, $G_{3,2}$ ; $G_{4,4}$, $G_{4,5}$ ; $G_{5,1}$, $G_{5,2}$,$G_{5,3}$, $G_{5,4}$, $G_{5,5}$.

The following tables gives the number of $HSC^n_{G_{d,r}}$ transitive groups of degree $\le 30$ related to the primitive group $G_{d,r}$, with $n>1$, and the first examples.

$\begin{array}{c|c} &G_{2,1}&G_{3,1}&G_{3,2}&G_{4,4}&G_{4,5}&G_{5,1}&G_{5,2}&G_{5,3}&G_{5,4}&G_{5,5} \newline \hline 2 &2&4&9&18&30&8&15&31&1&5 \newline \hline 3 &15&152&476 \newline \hline 4 &597 \end{array} $

$\begin{array}{c|c} &G_{2,1}&G_{3,1}&G_{3,2}&G_{4,4}&G_{4,5}&G_{5,1}&G_{5,2}&G_{5,3}&G_{5,4}&G_{5,5} \newline \hline 2 &G_{4,1}&G_{9,1}&G_{9,3}&G_{16,63}&G_{16,195}&G_{25,1}&G_{25,4}&G_{25,8}&G_{25,192}&G_{25,196} \newline \hline 3 &G_{8,1}&G_{27,1}&G_{27,8} \newline \hline 4 &G_{16,1} \end{array} $

These tables show the existence of all the $HSC^n_{G_{d,r}}$ transitive groups of degree $\le 30$.

**Remark**: There is one and only one $HSC^2_{G_{5,4}}$ transitive group: $G_{25,192}$, of order $46656000000$, whereas $G_{5,4} \simeq A_5$ is order $60$.

**p-adic philosophy for subfactors** (optional part)

Let $R$ be the hyperfinite II$_1$ factor, then $(R \subset R \rtimes \mathbb{Z}/ p^{n}\mathbb{Z})$ is a $HSC^n_{(R \subset R \rtimes \mathbb{Z}/ p\mathbb{Z})}$ subfactor.

We obtain the infinite chain $(R \subset R \rtimes \mathbb{Z}/ p\mathbb{Z} \subset R \rtimes \mathbb{Z}/ p^{2}\mathbb{Z} \subset \dots \subset R \rtimes \mathbb{Z}/ p^{n}\mathbb{Z} \subset \dots )$

Let $\mathbb{Z}_p$ be the group of p-adic integers.

Question: Is $(R \subset \bigcup_n R \rtimes \mathbb{Z}/ p^{n}\mathbb{Z}) \simeq (R \subset R \rtimes \mathbb{Z}_p)$ ?

**Remark**: $\mathbb{Z}_p$ is not a $HSC^{\infty}_{\mathbb{Z}/ p\mathbb{Z}}$ group because $m\mathbb{Z} \subset \mathbb{Z} \subset \mathbb{Z}_p$, for all $m \in \mathbb{N}$.

If there is a *generic* way to do this construction in the subfactor framework, we obtain that from the single maximal subfactor $(R \subset R \rtimes \mathbb{Z}/ p\mathbb{Z})$ we generate all the subfactors $(R \subset R \rtimes \mathbb{Z}/ m\mathbb{Z})$ as intermediate subfactors $(R \rtimes m\mathbb{Z} \subset R \rtimes \mathbb{Z})$ of $(R \subset R \rtimes \mathbb{Z}_p)$.

**Generalization**:

Now let $(P \subset Q)$ be a (finite index) maximal subfactor (for example the Haagerup subfactor), and suppose that there is a *canonical* $HSC^n_{(P \subset Q)}$ subfactor $(P \subset Q^{(n)})$ and an infinite chain $(P \subset Q^{(1)} \subset \dots \subset Q^{(n)} \subset \dots )$.

Let's call $(P \subset Q^{(\infty)})$ the $(P \subset Q)$-adic subfactor (with $Q^{(\infty)} = \bigcup_n Q^{(n)}$).

Classificationof the (finite index) intermediate subfactors of $(P \subset Q^{(\infty)})$ ?

Are they allnatural subfactors(see the optional part of this post).