Previously I explained how Auschwitz became an extermination camp: https://www.reddit.com/r/AskHistorians/comments/cvyw3n/what_was_the_situation_that_caused_germany_to/ey95h12/
Why some people are questioning Auschwitz is related to why some people are still questioning the shape of the Earth: those people are mostly ideology-driven and irrational, most of them possess pretty low intelligence too.
Indeed, even ignoring all the testimonies by the victims, perpetrators and the bystanders, the gas chambers, the mass murder and the burning pyres of Auschwitz are documented, so we know what it was during the WWII:
Key Auschwitz gas chambers documents: http://holocaustcontroversies.blogspot.com/2015/05/rebuttal-of-mattogno-on-auschwitz-part_3.html
The Franke-Gricksch report, describing the mass murder of Jews in the Auschwitz gas chambers: http://holocaustcontroversies.blogspot.com/2019/08/nazi-document-on-mass-extermination-of.html
More documents about mass murder of Jews in Auschwitz:
http://holocaustcontroversies.blogspot.com/2017/07/the-kinna-report-german-document-on.html
http://holocaustcontroversies.blogspot.com/2017/02/separate-accommodation-in-auschwitz.html
Photos of burning pyres as evidence for mass extermination:
http://holocaustcontroversies.blogspot.com/2012/08/the-auschwitz-open-air-incineration.html
Most Auschwitz extermination-related documents: http://holocaustcontroversies.blogspot.com/2012/10/index-of-published-evidence-on.html
As to the facilities, the question is a bit baffling: why wouldn't they have? To borrow a few paragraphs from something I wrote previously against the Holocaust deniers: Auschwitz consisted of numerous sub-camps and served many functions, among them: a POW camp, a labor camp, a detention camp, an extermination camp. There were tens of thousands of non-Jewish inmates in Auschwitz and its sub-camps too, who as a rule were subjected to a different treatment than Jews. And of course there were thousands of SS men who had to be entertained in their free time. Therefore there is no contradiction whatsoever between the camp having facilities the laymen may not associate with an extermination camp and the Holocaust.
Moreover, the Nazi extermination plan was never "let's kill every single Jew immediately". It was killing all Jews eventually. The Jews thought to be permanently unable to work due to sickness or age (in practice it was the majority) were gassed as a rule. But many of them, those able to work, were thought of as a valuable resource for the war economy, especially during the periods of labor shortage. Since every person eventually becomes unfit for work - sooner than later when it's back-breaking slave labor - all Jews were eventually to perish. But until then most Jews that the Nazis deemed "fit for work" could stay in the camp and even visit hospitals if it was thought that they would soon recover.
If some event took place in any of those subcamps, it could be said to have taken place "in Auschwitz", but the actual extermination camp was Birkenau. For a relatively short time there were also gassings in the main camp - Auschwitz I - which had a small makeshift gas chamber (currently shown to the tourists) where probably no more than 10000 people were murdered. But the extermination camp proper was Birkenau.
So the denier logic goes: if some innocent activity took place somewhere - anywhere - "in Auschwitz" - e.g. in any of its numerous subcamps, no matter how unrelated to and distant from the direct process of murder, it disproves that extermination took place where claimed. Does that logic make any sort of sense? Obviously not.
So the first question to ask is: where were the events or the facilities situated?
Moreover, Auschwitz obviously wasn't an exclusively Jewish camp, and Jews were treated differently than the other groups. And the thousands of the SS troops also needed recreational facilities and events.
So the second question to ask is: for whom were these facilities and events?
Finally, the mass extermination of Jews in Birkenau began in 1942.
So the third version is to ask: when did the facilities exist/were built or when did the events take place?
Let's take the swimming pool, for example. First of all it was built in 1944 in the main camp Auschwitz I (which was not an extermination site for Jews at that point in time). Second, who said that when used as a swimming pool, it was for the Jews and not for the non-Jewish prisoners, like Poles and Germans?
(Note: its actual function was to serve as a firefighting water reservoir, with the diving board added for a secondary use as a swimming pool; on the date of construction see e.g. Rudolf Kauer, Der 1. Frankfurter Auschwitz-Prozeß, 06.07.1964, S. 11723, 11770; on the use as a firefighting reservoir see e.g. the Dejaco-Ertl trial, Eugeniusz Nosal's statement on 09.02.1972 and Erwin Olszowka's statement on 11.02.1972; also note: the deniers claim the Soviets and the Poles were allegedly falsifying things all over Auschwitz; yet the pool is still there...)
There was a brothel in the Auschwitz main camp strictly for the non-Jewish prisoners (and of course with strictly non-Jewish women; "race-mixing").
There was a sports association strictly for the Germans on which Pressac comments:
As for the fencing competition, the swastika on the flag and the badge worn by the fencer on the right, the presence among the spectators of SS and SA officers, members of the party in uniform and of policemen, show that this is a meeting for “Reichsdeutschen”, exclusively for "sound" elements and no others.
Indeed, it was a sports association run by IG Auschwitz/IG Farben in Monowitz (Auschwitz III), quite away from the extermination sites. For example on 23.07.1944 it organized an athletics competition to which "the whole German staff of the construction site" was invited. Lying deniers of course would like people to believe that the sports association was for the prisoners. (It is not the only attempt by the deniers to present images made in Monowitz as if they were representative of Auschwitz as a whole; Pressac commented on such attempts, though he did not know the photos come from Monowitz, specifically from Dürrfeld's defense exhibit 133 at the IG Farben trial; see Dürrfeld's comments on each individual photo on 16.04.1948 in the IG Farben trial transcript, pp. 11619-11634).
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