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Carl-Fredrik Nyberg Brodda
Carl-Fredrik Nyberg Brodda
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Here is a very classical example. As stated in the comments, Gromov was an early proponent of importing ideas from geometry to group theory, but already thirty years earlier there was work in this direction. One of the most fundamental papers on geometric group theory was written in 1955 by A. S. Schwarz [1]. In this paper the famous "Schwarz-Milnor Lemma" is proved; it relates the growth of balls in the universal cover of a compact Riemannian manifold $M$ to the growth of balls in the fundamental group $\pi_1(M)$, and says that they are essentially the same (one speaks of quasi-isometries in modern terminology; this is then the same as the comment made by PaulSiegel).

At the end, he gives some applications of this result. First, he states two results:

Theorem 1. The volume of balls in a simply connected Riemannian manifold whose curvature is $K \leq C$, where $C <0$, cannot grow slower than $e^r$.

Theorem 2. The volume of balls in a simply connected $n$-dimensional Riemannian manifold of non-positive curvature ($K \leq 0$) cannot grow slower than $r^n$.

In both cases, $r$ refers to the radius of the balls in question. He attributes Theorem 2 to follow from results by É. Cartan (1936). Schwarz then uses his main theorem to state the two analogous results for growth in fundamental groups, after observing that the volume of balls in the universal cover of a compact Riemannian manifold with abelian fundamental group of rank $n$ grows as $r^n$:

Theorem 1'. The fundamental group of a Riemannian manifold with negative curvature ($K<0$) cannot be abelian.

Theorem 2'. The fundamental group of an $n$-dimensional compact Riemannian manifold with non-positive curvature $(K \leq 0)$ cannot be abelian of rank $<n$.

Finding the paper is, if I recall right, a bit tricky, even the original Russian one. Albert Schwarz sent me a copy of the Russian, and I translated it into English. I can provide a copy of either, if needed.

${}$

[1] A. S. Schwarz, "The volume invariant of coverings", Dokl. Akad. Nauk SSSR 105:1, 1955.

[Note: sometimes the title of this article is translated as "A volume invariant of coverings". The original Russian title "Объемный инвариант накрывающих" has no article "a/the", as this does not exist in Russian. But one can infer that "the" is meant, as the paper is not about defining some new invariant, but rather about proving facts about an existing invariant; this invariant, is attributed in the paper to an earlier article by V. A. Efremovich.]

Here is a very classical example. As stated in the comments, Gromov was an early proponent of importing ideas from geometry to group theory, but already thirty years earlier there was work in this direction. One of the most fundamental papers on geometric group theory was written in 1955 by A. S. Schwarz [1]. In this paper the famous "Schwarz-Milnor Lemma" is proved; it relates the growth of balls in the universal cover of a compact Riemannian manifold $M$ to the growth of balls in the fundamental group $\pi_1(M)$, and says that they are essentially the same (one speaks of quasi-isometries in modern terminology; this is then the same as the comment made by PaulSiegel).

At the end, he gives some applications of this result. First, he states two results:

Theorem 1. The volume of balls in a simply connected Riemannian manifold whose curvature is $K \leq C$, where $C <0$, cannot grow slower than $e^r$.

Theorem 2. The volume of balls in a simply connected $n$-dimensional Riemannian manifold of non-positive curvature ($K \leq 0$) cannot grow slower than $r^n$.

In both cases, $r$ refers to the radius of the balls in question. He attributes Theorem 2 to follow from results by É. Cartan (1936). Schwarz then uses his main theorem to state the two analogous results for growth in fundamental groups, after observing that the volume of balls in the universal cover of a compact Riemannian manifold with abelian fundamental group of rank $n$ grows as $r^n$:

Theorem 1'. The fundamental group of a Riemannian manifold with negative curvature ($K<0$) cannot be abelian.

Theorem 2'. The fundamental group of an $n$-dimensional compact Riemannian manifold with non-positive curvature $(K \leq 0)$ cannot be abelian of rank $<n$.

Finding the paper is, if I recall right, a bit tricky, even the original Russian one. Albert Schwarz sent me a copy of the Russian, and I translated it into English. I can provide a copy of either, if needed.

${}$

[1] A. S. Schwarz, "The volume invariant of coverings", Dokl. Akad. Nauk SSSR 105:1, 1955.

[Note: sometimes the title of this article is translated as "A volume invariant of coverings". The original Russian title "Объемный инвариант накрывающих" has no article "a/the", as this does not exist in Russian. But one can infer that "the" is meant, as the paper is not about defining some new invariant, but rather about proving facts about an existing invariant, attributed in the paper to V. A. Efremovich.]

Here is a very classical example. As stated in the comments, Gromov was an early proponent of importing ideas from geometry to group theory, but already thirty years earlier there was work in this direction. One of the most fundamental papers on geometric group theory was written in 1955 by A. S. Schwarz [1]. In this paper the famous "Schwarz-Milnor Lemma" is proved; it relates the growth of balls in the universal cover of a compact Riemannian manifold $M$ to the growth of balls in the fundamental group $\pi_1(M)$, and says that they are essentially the same (one speaks of quasi-isometries in modern terminology; this is then the same as the comment made by PaulSiegel).

At the end, he gives some applications of this result. First, he states two results:

Theorem 1. The volume of balls in a simply connected Riemannian manifold whose curvature is $K \leq C$, where $C <0$, cannot grow slower than $e^r$.

Theorem 2. The volume of balls in a simply connected $n$-dimensional Riemannian manifold of non-positive curvature ($K \leq 0$) cannot grow slower than $r^n$.

In both cases, $r$ refers to the radius of the balls in question. He attributes Theorem 2 to follow from results by É. Cartan (1936). Schwarz then uses his main theorem to state the two analogous results for growth in fundamental groups, after observing that the volume of balls in the universal cover of a compact Riemannian manifold with abelian fundamental group of rank $n$ grows as $r^n$:

Theorem 1'. The fundamental group of a Riemannian manifold with negative curvature ($K<0$) cannot be abelian.

Theorem 2'. The fundamental group of an $n$-dimensional compact Riemannian manifold with non-positive curvature $(K \leq 0)$ cannot be abelian of rank $<n$.

Finding the paper is, if I recall right, a bit tricky, even the original Russian one. Albert Schwarz sent me a copy of the Russian, and I translated it into English. I can provide a copy of either, if needed.

${}$

[1] A. S. Schwarz, "The volume invariant of coverings", Dokl. Akad. Nauk SSSR 105:1, 1955.

[Note: sometimes the title of this article is translated as "A volume invariant of coverings". The original Russian title "Объемный инвариант накрывающих" has no article "a/the", as this does not exist in Russian. But one can infer that "the" is meant, as the paper is not about defining some new invariant, but rather about proving facts about an existing invariant; this invariant is attributed in the paper to an earlier article by V. A. Efremovich.]

Source Link
Carl-Fredrik Nyberg Brodda
Carl-Fredrik Nyberg Brodda
  • 6.5k
  • 4
  • 27
  • 62

Here is a very classical example. As stated in the comments, Gromov was an early proponent of importing ideas from geometry to group theory, but already thirty years earlier there was work in this direction. One of the most fundamental papers on geometric group theory was written in 1955 by A. S. Schwarz [1]. In this paper the famous "Schwarz-Milnor Lemma" is proved; it relates the growth of balls in the universal cover of a compact Riemannian manifold $M$ to the growth of balls in the fundamental group $\pi_1(M)$, and says that they are essentially the same (one speaks of quasi-isometries in modern terminology; this is then the same as the comment made by PaulSiegel).

At the end, he gives some applications of this result. First, he states two results:

Theorem 1. The volume of balls in a simply connected Riemannian manifold whose curvature is $K \leq C$, where $C <0$, cannot grow slower than $e^r$.

Theorem 2. The volume of balls in a simply connected $n$-dimensional Riemannian manifold of non-positive curvature ($K \leq 0$) cannot grow slower than $r^n$.

In both cases, $r$ refers to the radius of the balls in question. He attributes Theorem 2 to follow from results by É. Cartan (1936). Schwarz then uses his main theorem to state the two analogous results for growth in fundamental groups, after observing that the volume of balls in the universal cover of a compact Riemannian manifold with abelian fundamental group of rank $n$ grows as $r^n$:

Theorem 1'. The fundamental group of a Riemannian manifold with negative curvature ($K<0$) cannot be abelian.

Theorem 2'. The fundamental group of an $n$-dimensional compact Riemannian manifold with non-positive curvature $(K \leq 0)$ cannot be abelian of rank $<n$.

Finding the paper is, if I recall right, a bit tricky, even the original Russian one. Albert Schwarz sent me a copy of the Russian, and I translated it into English. I can provide a copy of either, if needed.

${}$

[1] A. S. Schwarz, "The volume invariant of coverings", Dokl. Akad. Nauk SSSR 105:1, 1955.

[Note: sometimes the title of this article is translated as "A volume invariant of coverings". The original Russian title "Объемный инвариант накрывающих" has no article "a/the", as this does not exist in Russian. But one can infer that "the" is meant, as the paper is not about defining some new invariant, but rather about proving facts about an existing invariant, attributed in the paper to V. A. Efremovich.]