Is it true that the fundamental group of a compact finite dimensional Alexandrov space with curvature bounded below is finitely generated?


1 Answer 1


Under these hypotheses the systole of $X$ is clearly bounded from below, and the usual comparison arguments would give an upper bound on the number of points in an $\epsilon$-net in $X$. Then every loop can be discretized and the finite-generation follows.

  • $\begingroup$ Thank you very much. May I ask in addition whether the stronger statement is true: the minimal number of generators of the fundamental group is bounded above by a constant depending only on dimension, diameter, and a lower bound on curvature? $\endgroup$
    – asv
    May 10, 2018 at 5:47
  • 1
    $\begingroup$ I am not sure. You might want to try to run this by Anton Petrunin. $\endgroup$ May 10, 2018 at 8:57
  • $\begingroup$ It is possible that this would follow from a 2017 publication by Sabourau where he provides a specific lower bound for the systole, modulo suitable hypotheses on $X$. @orbits $\endgroup$ May 10, 2018 at 11:32

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