At the end of "The Red Circle," Holmes calls Emilia Lucca out of hiding and she sees that she is finally safe from Giuseppe Gorgiano of the Red Circle, a brotherhood allied to the old Carbonari. Through Mrs. Lucca's backstory, we learn that the Red Circle was a society well-versed in blackmail, violence, and murder.
Inspector Gregson takes Mrs. Lucca to see his chief to check out her story, the American Pinkerton agent is satisfied that his prey is dead, Holmes and Watson head off to the opera, and everyone lives happily ever after.
Unless you were one of the nameless individuals terrorized by this society.
Holmes showed no interest in addressing this organization and their crimes. One could imagine that they were a deep organizing power that pervaded society, made up of numerous and organized agents. We've seen Holmes take on a force like this before, the Moriarty gang.
So why did Holmes dismiss this organization? Was it because Gorgiano was the only active member in London and they did not threaten Holmes's city? I disagree with that.
In "The Six Napoleons," we hear of Pietro Venucci, one of the greatest cut-throats in London. And according to Lestrade, Venucci is connected with the Mafia.
Most chronologists agree that SIXN takes place in 1900, but the dates for REDC are all over the map, ranging between 1893 and 1903, so SIXN would be, at most, a few years off from REDC. And a few years would be plenty of time for a criminal organization to create a stronghold in London, especially after the hole left once Moriarty's organization crumpled in 1891. So the dates work.
But the word "Mafia" is never mentioned in REDC, and it's cavalier to paint all Italian criminals with such a broad brush.
So let's look at the ORIGINAL texts.
Some changes were made to the manuscripts before they saw publication. In REDC, "The Red Circle" was originally called "The Black Hand" and it's allegiant organization "the old Carbonari" was originally "the famous Camorra." Both of these original names were true-to-life criminal organizations at the time. In SIXN, Pietro Venucci was said to be from Florence. But in the original text, he was from Naples. The organization Watson called the Red Circle was also located in Naples.
I think we have at least enough evidence to argue that an Italian criminal organization was working in London around the turn of the century. So why would Sherlock Holmes walk away from both of these cases without further investigation into the larger society connected to them?
Maybe he did, and for whatever reason those facts were withheld from us. Maybe Holmes recognized that this organization spanned continents and wasn't one that he could address on his own. Or maybe somebody made him an offer he couldn't refuse.