But in terms of both crime and terrorism, immigrants are not the problem Trump says they are.

Study after study after study bears out mostly the last part of his remarks: Immigrants largely commit crimes at a lower rate than the local-born population. Those numbers are true even of the children of immigrants. Writing in the Oxford Handbook of Crime and Criminal Justice, Sandra M. Bucerius, an associate professor at the University of Alberta, noted:

“Second-generation immigrants typically have higher crime rates than first-generation immigrants. In the US context, however, most second-generation immigrants continue to enjoy lower crime rates than the native-born population. In stark contrast, research findings in European countries indicate that some second-generation immigrant groups have crime rates that drastically exceed those of the native-born population.”

I asked Bucerius about immigrants and crime (not terrorism), and she told me that though most studies do not differentiate among different immigrant groups, researchers do know there are some “immigrant groups in every Western country that we have data on that [are] more criminally involved than the average.”

“We also know,” she said, “that those groups always experience social, economic and/or political exclusion higher than the average. This does not imply that all immigrants who are socially excluded become criminals. Yet, exclusion and discrimination seem to be a risk factor.”

Bucerius points out that that while studies point broadly toward lower crime rates among immigrants and their children, these studies do not—and often cannot—speak to differences across different ethnic or religious groups. For instance, she says, there are no studies that compare crime rates among Muslims, Jews, and Christians.

“What we can say is that—in some European countries— like Germany, the Netherlands, Belgium, or France, we see a disproportionate number of second-generation immigrants involved in crime who are likely Muslim,” she said. “However, for these groups in particular, we need to take the very problematic history of guest-worker integration into account, and consider the highly problematic relationship between France and Algeria [and] forced secularism.”

Bucerius spent five years studying predominantly Muslim Turkish, Moroccan, and Albanian drug dealers in Germany, and the resulting book—Unwanted: Muslim Immigrants, Dignity and Drug Dealing—examines how different policies and exclusion practices cause and foster alienation among immigrants communities. For instance, she points out, most second-generation immigrants born in German in the 1970s, ’80s, ’90s are not German citizens.

“They were born and raised in a country that they could never become citizens of, and constantly live with the fear of deportation,” she said.