To what extent can global terrorism trigger local extremist crime, and among which groups? We draw on the full universe of police-recorded extremist crime in Germany in the years 2010 to 2020 to answer this question. Our data comprises 400,000 entries of three types of extremist violence: right-wing, left-wing, and Islamist. We connect these data with quasi-random variation in the occurrence of large-scale terrorist attacks around the world. Our analyses show that global terrorist attacks, mostly by Islamist terrorists, reliably predict local right-wing `revenge' attacks against foreigners, and also appear to inspire local Islamist violence. No effect shows for left-wing extremism. Robustness and falsification tests demonstrate that this relationship is likely causal. Further analysis show that the entire effect is mediated by broadband access. Global terrorism only triggers local extremist crime in areas with fast internet. By making distant violent terrorist attacks immediately emotionally relevant, the internet works as an amplifier, making global terrorism even more harmful than it typically atrocious immediate effects would suggest.