The White House press office did not respond to multiple requests for comment.
Right-wing extremists — a catchall category for a messy constellation of neo-Nazis, white nationalists, crypto-fascists, nihilists and attention-seeking trolls — vary widely in style and ideology. Some congregate out in the open, on forums like 4chan and Reddit as well as public platforms like Gab, the Twitter-like social network used by the suspect in the Pittsburgh synagogue shooting. Others communicate in private channels on Discord, a chat platform, or over encrypted messaging apps like Telegram or Wire. Some are ardent supporters of Mr. Trump, while others oppose him on the grounds that he is not extreme enough.
What they have in common is a feeling of empowerment — a sense that the boundaries of acceptable speech are widening in the Trump era, and a suspicion that when they talk, Mr. Trump, or those with access to him, may be listening.
Even small phrases can set off speculation. Last month, when Mr. Trump tweeted an unfounded accusation that left-wing protesters outside the confirmation hearings for Justice Brett M. Kavanaugh were “paid for by Soros and others,” some extremists took it as evidence that the president shared their view of a global Jewish-led conspiracy led by Mr. Soros, a leading donor to many liberal causes.
“Trump has officially named the Jew,” wrote one user on 4chan. “Trump knows,” wrote another, who said that the “others” Mr. Trump referred to in his tweet might be a sly reference to other shadowy Jewish benefactors.
These extremists’ sense of influence is almost certainly exaggerated. Mr. Trump, who has Jewish family members including his daughter, has said nothing about Mr. Soros’s religion or a larger Jewish conspiracy. And it is unlikely that the president, who has said that he does not use a computer, is wading through obscure message boards in search of talking points.
But that hasn’t stopped those extremists from interpreting his words as a signal that the president shares their views.
It is difficult to quantify how many right-wing extremists exist in America — many operate anonymously or pseudonymously online, and few real-world gatherings take place. But some private channels for neo-Nazis and other extremist groups have thousands of members, and more mainstream right-wing spaces — such as a pro-Trump Reddit forum, r/the_donald, which has more than 600,000 members — have amplified extremist messages.