Can I sue a property management company if they failed to evict a dangerous tenant and I was later injured by that tenant?

You *may* be able to sue. Generally, a landlord is not responsilble or liable for the criminal or negligent acts of its tenants: they are third-parties not under the landlord's direct control. But landlords do have a duty to take reasonable steps to protect tenants from foreseeable or predictible threats or risks, and this can include to protect against other tenants known to be violent and/or who have previously threatened the later-injured tenant.
For the landlord to be liable, several things are necessary:
1) The landlord must have been put on notice of the threat/risk--this is important, since if the assaut on you was the first criminal or disorderly act by this tenant, they landlord had no prior warning, therefore no reason to act, and so would not be liable; 
2) The landlord must have had a reasonable amount of time or opportunity to take action, which depends on the timing between when the landlord first became aware of the threat posed by this tenant and when you were injured; and
3) The landlord must have have had grounds to evict, since if the landlord could not evict, he could do nothing and so would not be liable--people are not liable for things outside their control.
I practice landlord-tenant law in NJ; since landlord-tenant law varies state by state, the following is offered as an example, only.
In NJ, a landlord generally has power to evict for disorderly, disruptive, or criminal acts on its own propety (this is an oversimpliciation, since the landlord's rights and power vary by whether it is a subsidized, like by section 8, or unsubsidized tenancy, by who the target of the disorderly, etc. act was, and in some cases by whether the act was  one-time occurence or repeated after the tenant was warned to not do it). However, they have limited or no power to act when the criminal or disorderly conduct occurs off or outside the landlord's property. Thus, if the assault, the shooting, the etc. occured offsite, the landlord would have no power to act and thus no liability, even if the landlord had sufficient warning to have acted before you were injured had the acts/events occured on the landlord's property. The landlord must have been able to take action to potentially be liable.
So all we can say that you *may* have grounds to sue, but whether you can depends on the specific facts and the timeline of events (e.g. was the assault on you this tenant's first bad act that the landlord would have been made aware of?). It would be worthwhile for you to consult in detail about your situation with local attorney, to see if under the facts of your situation, you do have a viable case or claim. Good luck.

You *may* be able to sue. Generally, a landlord is not responsilble or liable for the criminal or negligent acts of its tenants: they are third-parties not under the landlord's direct control. But landlords do have a duty to take reasonable steps to protect tenants from foreseeable or predictible threats or risks, and this can include to protect against other tenants known to be violent and/or who have previously threatened the later-injured tenant.
For the landlord to be liable, several things are necessary:
1) The landlord must have been put on notice of the threat/risk--this is important, since if the assaut on you was the first criminal or disorderly act by this tenant, they landlord had no prior warning, therefore no reason to act, and so would not be liable; 
2) The landlord must have had a reasonable amount of time or opportunity to take action, which depends on the timing between when the landlord first became aware of the threat posed by this tenant and when you were injured; and
3) The landlord must have have had grounds to evict, since if the landlord could not evict, he could do nothing and so would not be liable--people are not liable for things outside their control.
I practice landlord-tenant law in NJ; since landlord-tenant law varies state by state, the following is offered as an example, only.
In NJ, a landlord generally has power to evict for disorderly, disruptive, or criminal acts on its own propety (this is an oversimpliciation, since the landlord's rights and power vary by whether it is a subsidized, like by section 8, or unsubsidized tenancy, by who the target of the disorderly, etc. act was, and in some cases by whether the act was  one-time occurence or repeated after the tenant was warned to not do it). However, they have limited or no power to act when the criminal or disorderly conduct occurs off or outside the landlord's property. Thus, if the assault, the shooting, the etc. occured offsite, the landlord would have no power to act and thus no liability, even if the landlord had sufficient warning to have acted before you were injured had the acts/events occured on the landlord's property. The landlord must have been able to take action to potentially be liable.
So all we can say that you *may* have grounds to sue, but whether you can depends on the specific facts and the timeline of events (e.g. was the assault on you this tenant's first bad act that the landlord would have been made aware of?). It would be worthwhile for you to consult in detail about your situation with local attorney, to see if under the facts of your situation, you do have a viable case or claim. Good luck.