I believe it is common knowledge that American police are allowed lie to a suspect, but how far can they go?
Can they say / show a suspect things like these examples--
We have a witness who saw you steal the car? (They don't)
We have a written statement from the witness? (They don't)
Here is the witness' written statement. (It's a fake document the police made up)
Here is a video of you stealing the car (It's a deep fake video the police made up)
For the most part, U.S. police are allowed to lie to suspects to obtain information, even if they are not undercover. This goes pretty far although the exact boundaries are a matter of case law and are very fact sensitive. The factors missing from the examples in the question are the details about the suspect, whose sophistication, mental capacity, and the degree to which the false statements are equivalent to duress under the "totality of the circumstances" are relevant.
Facts similar to the first two examples have not been sufficient to suppress confessions obtained with them. In a 1989 case from Florida, something similar to the third example was sufficient to invalidate a confession. To my knowledge, there are no examples, yet, of the fourth "deep fake" type of deception being used in a precedent establishing case.
As explained here:
When people are brought in for questioning by the police, they are expected to tell the truth or get in trouble. But what about the officers questioning you — are they bound to be truthful?
In Colorado and most states, the short answer is no. They can’t lie in every instance, and they can’t fabricate evidence (Florida v. Cayward, 1989), but most of the time it’s completely legal for them to lie so it’s important to remember this if you are ever interrogated. The Supreme Court ruled in Frazier v. Cupo (1969) that police officers can lie during an investigation as long as it does not “shock the conscience of the court or the community.” For instance, they can’t tell someone that they will lose custody of their children if they don’t confess (Lynumn v. Illinois, 372 US 528 (1963). But they most certainly can lie about a lot of things that can intimidate people into confessing to crimes they didn’t commit. And as long as the court determines that a confession was voluntary and not obtained through violence, the court could allow it.
Here are some relevant details:
In Cayward, the Florida court of appeals held that the defendant's confession was inadmissible when a police officer fabricated two forensic laboratory reports and presented them to the defendant during the interrogation session, after which the defendant confessed.
In Frazier v. Cupp, 394 U.S. 731 (1969):
Petitioner in this habeas corpus proceeding claims that his constitutional rights were violated in three respects in his trial in 1965 for murder for which he had been indicted jointly with one Rawls, who pleaded guilty. (1) The prosecutor, on the basis of previous information he had received that Rawls would testify, included in his opening statement a brief summary of Rawls' expected testimony. When Rawls was called to the stand, he claimed his privilege against self-incrimination, and was dismissed. Petitioner's motions for a mistrial were overruled. The trial court instructed the jury that the opening statements of counsel should not be considered as evidence. (2) After preliminary questioning shortly after his arrest, petitioner was told that he could have an attorney if he wanted one and that anything he said could be used against him at trial. Thereafter the interrogating officer falsely told petitioner, who was reluctant to talk, that Rawls had confessed. Petitioner later began to spill his story, but again showed signs of reluctance and said he thought he better get a lawyer before he talked any more. Following the officer's reply that petitioner could not be in any more trouble than he was in, petitioner fully confessed and, after further warnings, signed a written confession, which was later admitted into evidence over petitioner's objection. . . . Petitioner was convicted, and the State Supreme Court affirmed. Petitioner thereafter filed a petition for a writ of habeas corpus in the District Court, which granted the writ. The Court of Appeals reversed. Petitioner claims that the prosecutor's use of the summarized Rawls statement denied him his constitutional rights of confrontation as guaranteed by the Sixth and Fourteenth Amendments; that his confession contravened the principles established by Escobedo v. Illinois, 378 U. S. 478 (1964), and Miranda v. Arizona, 384 U. S. 436 (1966), and was involuntary, and that the clothing had been illegally seized in violation of the Fourth and Fourteenth Amendments.
Held:
On the facts here, where the evidence which the prosecutor reasonably expected to produce was objectively and briefly summarized and was not touted to the jury as crucial to the prosecution's case, the court's limiting instructions were sufficient to protect petitioner's constitutional rights. Douglas v. Alabama, 380 U. S. 415 (1965), and Bruton v. United States, 391 U. S. 123 (1968), distinguished. Pp. 394 U. S. 734-737.
In the context of this case, where it is possible that the questioning officer took petitioner's remark about seeing an attorney not as a request that the interrogation cease but as a passing comment, there was no denial of the right to counsel such as existed in Escobedo, and Miranda, which was decided after petitioner's trial, is inapplicable under Johnson v. New Jersey, 384 U. S. 719 (1966). Pp. 394 U. S. 738-739.
On the facts of this case and in view of the "totality of the circumstances," the trial court did not err in holding that petitioner's confession was voluntary. P. 394 U. S. 739. . . .
This is controversial in part because:
Judicial permission for the police to use deceptive practices in investigations (undercover work, 'sting' operations, etc.) often carries over into the interrogation and testimony phases of the police development of a case, as police become accustomed to the use of deception to gain the conviction of a defendant they consider guilty.
It is particularly controversial in the case of juveniles, and some states (including Colorado and Washington State) have considered banning the use of deception in interrogations of juveniles, where the risk of a falsely obtained confession is perceived as being particularly great.
Incidentally, the same is not true of prosecuting attorneys. It is a violation of professional ethics for a prosecuting attorney to lie, even to reduce harm in a hostage situation, as one Colorado professional ethics case where this happened established.
Any confession made subsequent to those lies is at risk of being inadmissible. However, no particular lie, taken out of context, necessarily leads to a confession being rendered inadmissible. The assessment is contextual, where the lie is considered in combination with all other factors.
See R. v. Oickle, 2000 SCC 38 (internal citations removed).
The test is:
whether the police improperly induced the [defendant]’s confessions through threats or promises, an atmosphere of oppression, or any other tactics that could raise a reasonable doubt as to the voluntariness of his confessions.
...
Oppression clearly has the potential to produce false confessions. If the police create conditions distasteful enough, it should be no surprise that the suspect would make a stress-compliant confession to escape those conditions. Alternately, oppressive circumstances could overbear the suspect’s will to the point that he or she comes to doubt his or her own memory, believes the relentless accusations made by the police, and gives an induced confession.
...
A final possible source of oppressive conditions is the police use of non-existent evidence. As the discussion of false confessions revealed, this ploy is very dangerous. The use of false evidence is often crucial in convincing the suspect that protestations of innocence, even if true, are futile. I do not mean to suggest in any way that, standing alone, confronting the suspect with inadmissible or even fabricated evidence is necessarily grounds for excluding a statement. However, when combined with other factors, it is certainly a relevant consideration in determining on a voir dire whether a confession was voluntary.
...
the confession should be excluded if the police deception shocks the community. Second, even if not rising to that level, the use of deception is a relevant factor in the overall voluntariness analysis. At this stage, the approach is similar to the one used with fabricated evidence, though of course the use of inadmissible evidence is inherently less problematic than fabricated evidence.
