In the paper of Bannon and Ravichandran, A Folner invariant for type $\rm{II}_1$ factors, they defined an invariant $Fol(M)$ for a separable type $\rm{II}_1$ factor $M$, especially for the free group factor $M=L(F_m)$, where $F_m$ is the non-abelian free group on $m$ generators $\{a_1, \cdots, a_m\}$, by measuring how badly $M$ fails to satisfy Connes' Folner-type condition.

Let us focus on the pre-invariant $Fol(L(F_m), X_m)$, where $X_m=\{U_{a_1},\cdots, U_{a_m}\}$.

Here $U_{a_l}: l^2(F_m)\to l^2(F_m)$ are the unitary operators defined by $U_{a_l}(\sum_{g\in F_m}x_g\delta_g)=\sum_{g\in F_m}(x_g\delta_{a_lg})$, for any $\sum_{g\in F_m}x_g\delta_g\in l^2(F_m)$.

Note that $\delta_g\in l^2(F_n)$ denotes the characteristic function on the word $g$.

In their paper, they defined

$$Fol(L(F_m), X_m)=inf \lbrace \epsilon>0: Q(X_m,\epsilon) \rbrace$$

Here, we write $Q(X_m, \epsilon)$ to denote the following property holds:

There exists a nonzero finite-rank, say rank $k$, projection $e\in B(l^2(F_m))$ such that $\forall l\in\{1, 2, \cdots, m\}$,

$$\sum_{i,j=1}^k|T^l_{i,j}|^2\geq k(1-\frac{\epsilon^2}{2})$$ and $$|\sum_{i=1}^kT^l_{i,i}|\leq k\epsilon$$

where, $T^l:=eU_{a_l}e=(T^l_{i,j}), l=1,\cdots, m$ are the $k\times k$ matrices.

If we pick $\{\xi_1,\cdots, \xi_k\}$ to be the orthonormal basis with unit length of the dimension $k$ subspace $el^2(F_m)$, then note that $T^l_{i,j}=\langle eU_{a_l}e\xi_j,\xi_i \rangle=\langle U_{a_l}e\xi_j,e\xi_i \rangle=\langle U_{a_l}\xi_j,\xi_i \rangle$.

So, we can first pick any $k>0$ and any orthonormal basis with unit length $\xi_1,\cdots, \xi_k\in l^2(F_m)$, then calculate the $T^l$'s to find the possible value of $\epsilon$ such that $Q(X_m, \epsilon)$. Then, this would give us a upper bound for $Fol(L(F_m), X_m)$.

Now, my question is:

**What is the lower bound for $Fol(L(F_m), X_m)$? Does it depend on $m$?**

Known Results:

1, In Jon and Mohan's paper, they showed that $Fol(L(F_2), X_2)>0$.

2, Julien has given an elegant argument in http://math.stackexchange.com/questions/347786 , showing that for any $\delta>0$, $Q(X_2, \frac{\sqrt{3}}{2}+\delta)$ holds for some rank 1 projection, so $Fol(L(F_2), X_2)\leq \frac{\sqrt{3}}{2}$. In fact, if you follow the same argument of Julien for $F_m, m>2$, you find $Fol(L(F_m), X_m)\leq \sqrt{2}(\frac{m-1}{m})$. Note that we have obtained the upper bound that $Fol(L(F_m), X_m)\leq \sqrt{2(1-\frac{1}{m})}$ by constructing partitions of $F_m$.

Remarks:

1, Note that Julien's construction of the coefficient $x_g's$ for $\delta_g$ happens to be used in the well-known word-length deformation in Popa's deformation/rigidity theory. So, it would be nice to see whether this can be explored further.

2, It might be interesting to see whether we can use Julien's construction locally, i.e., use it for a subset of $F_2$, in other words, combine his construction with the partition of $F_2$, to get a better upper bound.