Denver saw a 2.2 percent drop in violent crime rates in the year after the first legal recreational cannabis sales in Colorado. Overall property crime dropped by 8.9 percent in the same period there, according to figures from the Drug Policy Alliance. In Washington, violent crime rates dropped by 10 percent from 2011 to 2014. Voters legalized recreational marijuana there in 2012.
Medical marijuana laws, which have a longer track record for academics than recreational pot legalization, are also associated with stable or falling violent crime rates.
Any violence associated with this particular drug is apparently connected with factors such as turf wars and deals gone bad that are common with the activity of trafficking the drug illegally. So in this case it isn't the drug itself causing violence, but rather the consequences of the illegal drug trade. This is something many of us who study aggression and violence from various disciplines would have known for ages.
Nor does experimental research suggest a link between marijuana intoxication and aggressive behavior in general. Note that in lab research, we don't measure violence, but we can measure physical aggression by having participants believe that they are delivering electric shock to another person. The higher the shock level, the more aggressively the participants behave. Myerscough and Taylor (1985) found evidence that participants who were given higher doses of TCH (the active ingredient in marijuana) were generally non-aggressive across provocation levels, suggesting that marijuana did not facilitate aggressive behavior. That followed up an earlier experiment in Taylor's lab suggesting that although alcohol intoxication facilitated aggression when participants were provoked, THC did not facilitate aggression under conditions of provocation.
More recently, Perna et al. (2016) showed that particpants who reported being either heavy alcohol or marijuana users showed an increase in aggression following alcohol intoxication and a decrease in aggression following marijuana intoxication. In addition, participants' subjective aggression was measured, showing that alcohol increased participants subjective aggression but that increase was not found among those in the marijuana condition. Perna et al. (2016) also provide a useful summary of the research literature, suggesting that much of the literature on marijuana and aggression (to the extent it exists) is largely hampered by very small samples, failure to include placebo conditions, and so on. The samples in the Perna et al (2016) experiment are also a bit smaller than I would want to see. However, as a generally well-designed exploratory experiment, it does suggest that we think twice before accepting the pronouncements of a politician on faith. The findings are also generally consistent with much of the earlier experimental and cross-sectional research that I am aware of.