I seek some help identifying a short book I read in my childbook. It was a science fiction book about a society where due to scarcity of resources, you were allowed a certain life span depending on your achievements and contributions to society. If you exceed the amount before doing something that causes you to get promoted to a higher life span, you will be taken in the night by some sort of police and vanish.
For extraordinary achievements the highest status you could achieve was immortality (I seem to dimly remember the usage of the word amaranth for those people).
The protagonist of the novel was someone who had once been granted immortaility but then had it stripped away from him again due to some misdeed or mistake. Now he tries to get that status again (maybe under a fake identity?), but one of the immortal people is suspicious of him so he murders her. She gets revived later, because that's part of the immortality package.
It ends with him being apprehended, but society decides to invest heavily into space again so that hopefully in the future there is space for everyone to be immortal. The book ends with the protagonists being sent of an explorer into space alone.