Scientific American offers a good summary of the evidence on this issue:
Immigration-crime research over the past 20 years has widely corroborated the conclusions of a number of early 20th-century presidential commissions that found no backing for the immigration-crime connection. Although there are always individual exceptions, the literature demonstrates that immigrants commit fewer crimes, on average, than native-born Americans.
Also, large cities with substantial immigrant populations have lower crime rates, on average, than those with minimal immigrant populations.Read the entire post (which was written by the researchers themselves, and not filtered by journalists).
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Across our studies, one finding remains clear: Cities and neighborhoods with greater concentrations of immigrants have lower rates of crime and violence, all else being equal.