What exactly IS the overpopulation argument (in regards to immortality)?

Periodically in discussions of cryonics and related issues, people bring up “the overpopulation argument, and the counterarguments that respond to it” without actually describing those arguments in detail. Overpopulation was a big concern of mine prior to exposure to anti-deathism. And so far, it still is. I have no philosophical problem with eliminating death, but it seems to me that getting rid of death BEFORE we eliminate all the other major problems facing humanity is going to make those problems much worse. Death is an enemy that should be vanquished, but it’s not an enemy I’m prepared to destroy until I’m satisfied that we’ll be able to handle the consequences.

If death is solved via uploads running at high speed, I’m not too concerned. (Still a little concerned, since computers still take up space, but the issue is close enough to negligible that I’m fine ignoring it).

If we’re dealing with physical humans taking up physical space requiring physical resources, then I’m worried. Either we’re growing exponentially, or we’ve eliminated childbirth, or we have strict rules in place about people who have children being willing to die. (The latter might work but modifying central aspects of the human life cycle that we are hard-wired to value seems.… challenging, to say the least)

The sense of I’ve gotten around here is that “exponential growth is okay, because Space is Big”. Space certainly is big, and I imagine we could expand for a long time without running into conflict. But if there’s even one other alien race who solves their problems the same way (expanding whenever they run out of space or resources), then eventually there’s going to be conflict. And the longer we go BEFORE that conflict, the more human suffering it might entail. I’d prefer to have achieved equilibrium as a species beforehand.

I want to expand into the universe, but I think we should do so out of *curiosity* rather than *necessity.*

I assume there’s been a lot of discussions about this and I don’t want to rehash them. But if someone could summarize the issues at hand and explain what the community consensus is, I’d appreciate it. (“Community Consensus” might mean “there are a few dominant schools of thought here”).

Edit: FAWS pointed out that as long as, on average, each person reproduces slightly less than once, growth will not continue exponentially. That’s pretty much the answer I’m looking for. The logistics are still significant, but in the long run I think such a law would be enforcible. (Each person only gets to reproduce once, and some people will choose not to. I think there’d need to be an additional disincentive, because over the course of an immortal lifespan, people are likely to try out childrearing at *some* point).

I still think that the issues surrounding overpopulation should occupy at least as much of our attention as ending death (basically ensuring that people are given adequate resources to live healthy, productive lives).