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Short story begins with the rescue of one person trying to escape a planet using a solar sail. He describes how his people go through population build-ups and collapses, endlessly.

The finish is always the same, all traces of civilization are destroyed, the last survivors fighting hand-to-hand. Evolution favored those with one arm shaped like a club or axe. This answers the rescuers question about his oddly nonsymmetric arms. They hurry to the planet in the hopes of stopping the cycle. The story was part of a collection in the late 60's, I believe.

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This is The Mote in God's Eye by Larry Niven and Jerry Pournelle, part of the latter's CoDominium series, and is actually a novel published in 1974. The aliens are called "Moties" because their home star appears to a nearby human colony as a speck in front of a red giant star, called Murcheson's Eye.

In the series, FTL travel happens by traveling to a particular point between two stars, then instantaneously jumping to the equivalent point near the other star. The Moties are trapped on their home planet because the only jump point available to them ends up inside the nearby red giant, where their ships get destroyed. The light-sail-powered craft is an attempt to escape the home planet by travelling to a different star nearby, which (fortunately) is already inhabited by humans.

The Moties are divided into several subspecies: we learn about Masters, Mediators, Engineers, Watchmakers and some others. The Moties' civilization is doomed to repeat the cycle of expansion and collapse because:

biologically, they have to reproduce or they will die, and competition between the different Masters ensures that the same mistakes (such as breeding Warrior Moties) will be reproduced on every cycle.

There is also a sequel, The Gripping Hand (published as The Moat around Murcheson's Eye in the UK), which deals with the Moties...

... escaping their home planet via a different route.

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While I think this is correct, note that it's not a perfect match for the book as described. The CoDominium expedition to the Mote is not sent with the aim of stopping the cycles -- they don't even learn about them until near the end of the mission (and book). –  Mike Scott May 18 at 16:27
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@MikeScott There are other differences too: the Crazy Eddie Probe's pilot was dead on capture, and it wasn't a short story. But the other plot points match so closely, that I'm certain it has to be The Mote in God's Eye. –  Niall C. May 18 at 17:04
    
I don't think it was the Mote, though there are a lot of similarities. In this one, just about the whole story is a narrative told by the person who was trying to escape from the planet, describing (to the people who found him) the cycles of collapse, buildup, collapse. IIRC it was the lead story in a short story collection –  puttster May 27 at 16:16
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