Donald Trump spoke about immigration in Phoenix on Wednesday, several hours after a joint press conference with Mexican President Enrique Peña Nieto. Photograph by David Paul Morris / Bloomberg via Getty

This essay is part of a series The New Yorker will be running through the election titled “Trump and the Truth.”

“The facts aren’t known because the media won’t report on them,” Donald Trump declared during his immigration speech in Phoenix on Wednesday. A few hours earlier, the Republican nominee had been in Mexico City, where he had held a joint press conference with the Mexican President, Enrique Peña Nieto, and lauded Mexican-Americans as “amazing people . . . just beyond reproach.” In Phoenix, flanked by American flags, he struck a different tone. Trump warned the crowd that if Clinton were elected, America would be inundated by a new wave of illegal immigration that would result in “thousands of more violent, horrible crimes, and total chaos and lawlessness.”

Again and again in his Presidential campaign, Trump has issued sweeping assertions about how immigrants are “bringing crime” to America. Wednesday offered only the latest, and loudest, example. Examining these claims is instructive, not for what they tell us about Trump but for what they reveal about immigrants, whose relationship to crime is greatly misunderstood. If you live in a city that has become less dangerous in recent decades, a growing body of evidence suggests that you actually have immigrants to thank.

When Trump kicked off his campaign, last year, he accused Mexico of sending “rapists” and criminals to America. This was a patently outrageous claim, and there was no evidence behind it. According to Robert Sampson, a sociologist at Harvard and the former scientific director of the Project on Human Development in Chicago Neighborhoods, communities with high concentrations of immigrants do not suffer from outsized levels of violence. The opposite is the case. In his exhaustively researched 2012 book, “Great American City,” Sampson noted that, in Chicago, “increases in immigration and language diversity over the decade of the 1990s predicted decreases in neighborhood homicide rates.” Other scholars have turned up similar findings. In 2013, a team of researchers published a paper on Los Angeles that found that “concentrations of immigrants in neighborhoods are linked to significant reductions in crime.” A 2014 study examining a hundred and fifty-seven metropolitan areas in the United States found that violent crime tended to decrease when the population of foreign-born residents rose.

One reason for this may be that immigrants have helped revitalize formerly desolate urban neighborhoods, starting businesses and lowering the prevalence of vacant buildings, where violence can take root. Another possible factor is that, contrary to Trump’s bigoted rhetoric, many immigrants are ambitious strivers, who are highly motivated to support their families and make a better life for their children. Undocumented immigrants may also be more fearful than legal residents of attracting police attention, which could get them deported. Whatever the answer, the research helps make sense of the fact that, during the nineteen-nineties, the foreign-born population in the U.S. grew by more than fifty per cent, and cities such as New York, El Paso, and San Diego, where many of these newcomers settled, did not become cauldrons of violence. They became safer. While numerous factors may have contributed to this—most notably, a stronger economy—it is now widely agreed that immigration played a role, too.

In his statements and speeches, Trump has often qualified his language by distinguishing between documented and undocumented immigrants. In Phoenix, he contrasted the “dangerous criminal aliens” who ruthlessly victimized Americans with legitimate, law-abiding immigrants who obeyed the rules. Data isolating the level of crime among undocumented immigrants are hard to come by, in part because, many suspect, undocumented-immigrant communities underreport crime, for fear of bringing law enforcement to their doors. Sampson acknowledged this in an e-mail to me recently, but he countered that one felony—murder—resists underreporting. And, in the nineteen-nineties, he noted, the murder rate fell dramatically in America’s cities, even as the flow of documented and undocumented immigrants surged. “The homicides committed by illegal aliens in the United States are reflected in the data just like for everyone else,” he wrote to me. “The bottom line is that as immigrants poured into the country, homicides plummeted.”

To point these things out is not to deny the tragedy of any individual killing or crime. In June, the Boston Globe_ published a deeply reported story showing that, among three hundred and twenty-three foreign-born criminals released in New England between 2008 and 2012, nearly one-third went on to commit new offenses after being released, including rape and attempted murder. Based on a three-year review of court records and police logs, the Globe _article raised legitimate questions about how forthright immigration officials have been about the issue. Trump cited the Globe’s story in his Phoenix speech, without acknowledging any of its nuance. Some of the criminals the Globe wrote about had entered the United States legally, and had been released because the government cannot detain immigrants indefinitely and because the countries they came from refused to take them back. The Globe also reported that, in the past year, almost sixty per cent of the people deported by Immigration and Customs Enforcement were convicted criminals. Trump skirted around these facts, assailing President Obama and Hillary Clinton for “surrendering the safety of the American people to open borders” and promising that the mayhem he described would end if he were elected. “The crime will stop,” he vowed.

On the few occasions when journalists have challenged him directly on his incendiary claims about immigration, Trump has resorted to bullying and denial. “If you look at the statistics of people coming . . . you look at the statistics on rape, on crime, on everything coming in illegally in this country, it’s mind-boggling!” Trump declared during an interview on CNN last year, citing as evidence a 2014 story published by Fusion that investigated the prevalence of sexual assaults against migrant women and girls en route to the United States. When Don Lemon, the CNN host, responded by pointing out that the Fusion story was about migrants who were raped, not hordes of criminals crossing the border, Trump snapped, “Well, somebody’s doing the raping, Don! . . . Who’s doing the raping?” A few days later, Trump was interviewed by the NBC News correspondent Katy Tur, who pointed out to him that the incarceration rate for both documented Mexican immigrants and undocumented immigrants over all was lower than that for native-born Americans. “It’s a wrong statistic,” Trump insisted. “Take a look at all the crime that’s being committed.” When Tur noted that the research “does not match what you’re saying,” Trump replied, “Don’t be naïve. You’re a very naïve person.” (One widely cited 2008 study from the Public Policy Institute of California found that the incarceration rate for foreign-born adults in the state prison system was two hundred and ninety-seven per hundred thousand in the population. For U.S.-born adults, the figure was eight hundred and thirteen per hundred thousand.)

When it comes to the threat that immigrants pose, Trump has championed perceptions over facts. Here, alas, he may be on to something. One of the more striking findings of Robert Sampson’s work has been that the “perceived disorder” of a neighborhood correlated positively with the prevalence of Latino residents, regardless of the actual level of crime. Trump certainly perceives disorder, and, with his false claims and invented statistics, he is contributing to the perception of it, especially among people who have few encounters with actual Hispanic immigrants. A recent study by a Gallup economist found that Trump’s supporters appear to be concentrated in segregated communities, far away from the Mexican border.

For more than a year, Trump has used the spotlight on him to smear and malign tens of millions of people who have made the communities they inhabit safer places for people of all backgrounds to raise children or pursue their dreams. They’ll be gone, he says—he’s building a wall. Americans deserve to be reminded that these ideas aren’t just frightening in their intolerance. If your concern is public safety, they’re backward.