In "The Adventure of the Dancing Man" by Conan Doyle, when Elsie gave Abe the slip in Chicago, how did Abe know she went to London?

The fact that Elsie was unable to give Abe the slip in spite of going all the way to London is part of what makes "The Adventure of the Dancing Men" seem so uncanny. The reader does not know how Abe Slaney and his gang could have traced her to England, but nevertheless the reader believes that it was done. The name Chicago in those days was synonymous with lawlessness and criminal power. When Slaney is captured, he does not offer to explain exactly how he had managed to trace Elsie to London. He simply tells his captors:

It was only after her marriage to this Englishman that I was able to find out where she was. I wrote to her, but got no answer. 

The letter he wrote to her would have been the one that, according to her husband, caused her such alarm when it was delivered to his estate. In their first interview, Mr. Hilton Cubitt tells Holmes:

One day my wife received a letter from America. I saw the American stamp. She turned deadly white, read the letter, and threw it into the fire. 

She was horrified when she realized that Slaney knew where to find her, regardless of her having fled all the way to England and her new last name.

"The Adventure of the Dancing Men" was first published in 1903. Foreign travel from America was much lighter in those days than it is today. Elsie would have had to buy her boat ticket in Chicago and then travel by train to New York to disembark. There would not have been many places in Chicago to purchase a boat ticket in 1903, and Abe Slaney with his gang could easily have checked all of them in a single day. It should not have been hard to trace a young woman traveling alone in those days. Elsie would have needed a passport, and the agency where she bought her boat ticket would probably have insisted on verifying her name via her passport. Slaney could have found out in Chicago that Elsie had traveled to London. Then he should have been able to trace her through the official British records or the London newspapers when she married Hilton Cubitt. It would not have been hard to trace Cubitt to his estate, since he was an important man in England.

At their first interview, Cubitt tells Holmes that he met Elsie at a boarding house in London, where they fell in love. Cubitt makes it clear that Elsie was living under her real name and not an assumed name when he says:

There was an American young lady there—Patrick was the name—Elsie Patrick. 

We know that this was her real name because Abe Slaney uses the name of Patrick when he is confessing after being caught.

There were seven of us in a gang in Chicago, and Elsie's father was the boss of the Joint. He was a clever man, was old Patrick.

Slaney, through British connections or private detectives, should have been able to locate a record of the marriage between Hilton Cubitt and Elsie Patrick, especially in Britain, where records go back for many centuries. Then it would have been easy to find Mr. Cubitt's address. Perhaps the author, Sir Arthur Conan Doyle, gave Holmes' client the unique name of Hilton Cubitt because that would have made him that much easier to locate.