Dynamics of foliations
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Foliations of Riemannian manifolds can be considered as multidimensional dynamical systems: Riemannian structure (more precisely, Riemannian structure along the leaves) plays the role of time, the leaves play the role of orbits \cite{Walczak2004}. The dynamics of a foliation can be described in terms of its holonomy (see [[Foliations#Holonomy]]) pseudogroup. As far as we know, the notion of a pseudogroup appeared for the | Foliations of Riemannian manifolds can be considered as multidimensional dynamical systems: Riemannian structure (more precisely, Riemannian structure along the leaves) plays the role of time, the leaves play the role of orbits \cite{Walczak2004}. The dynamics of a foliation can be described in terms of its holonomy (see [[Foliations#Holonomy]]) pseudogroup. As far as we know, the notion of a pseudogroup appeared for the | ||
− | first time in \cite{Veblen&Whitehead1932} in the context of geometric objects studied in differential geometry. Pseudogroups have been brought into the foliation theory by Haefliger \cite{Haefliger1962a}. The growth types of leaves (equivalent to the growth types of the corresponding orbits of holonomy pseudogroups) as well as the expansion growth of a foliation (or, its holonomy pseudogroup) describe some aspects of the foliation dynamics. Corresponding to the expansion growth is the notion of geometric entropy of a foliation \cite{Ghys&Langevin&Walczak1988}. Vanishing entropy can be seen as a dynamical condition which occurs to have strong geometric and topological consequences (see [[Dynamics of foliations#Results on entropy]]) below. | + | first time in \cite{Veblen&Whitehead1932} in the context of geometric objects studied in differential geometry. Pseudogroups have been brought into the foliation theory by Haefliger \cite{Haefliger1962a}. The growth types of leaves (equivalent to the growth types of the corresponding orbits of holonomy pseudogroups) as well as the expansion growth of a foliation (or, its holonomy pseudogroup) describe some aspects of the foliation dynamics. Corresponding to the expansion growth is the notion of geometric entropy of a foliation \cite{Ghys&Langevin&Walczak1988}. Vanishing entropy can be seen as a dynamical condition which occurs to have strong geometric and topological consequences (see [[Dynamics of foliations#Results on entropy]]) below.</wikitex> |
− | </wikitex> | + | |
− | + | ||
− | + | ||
== Pseudogroups == | == Pseudogroups == | ||
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\label{def:psgroup}{{endthm}} | \label{def:psgroup}{{endthm}} | ||
− | |||
As examples, one may have (1) all the homeomorphisms between open sets of a topological space, (2) all the diffeomorhisms of a given class (say, $C^k$, $k = 1, 2, \ldots, \infty, \omega$, between open sets of a manifold, (3) all the restrictions to open domains of elements of a given group $G$ of homeomorphisms of a given topological space. (In the last case, we say that the pseudogroup is '''generated''' by $G$.) | As examples, one may have (1) all the homeomorphisms between open sets of a topological space, (2) all the diffeomorhisms of a given class (say, $C^k$, $k = 1, 2, \ldots, \infty, \omega$, between open sets of a manifold, (3) all the restrictions to open domains of elements of a given group $G$ of homeomorphisms of a given topological space. (In the last case, we say that the pseudogroup is '''generated''' by $G$.) | ||
Any set $A$ of homeomorphisms bewteen open sets (with domains covering a space $X$) generates ma pseudogroup $\Gamma (A)$ which is the smallest pseudogroup containing $A$; precisely a homeomorphism $h:D_h\to R_h$ belongs to $\Gamma (A)$ if and only if for any point $x\in D_h$ there exist elements $h_1, \ldots, h_k\in A$, exponents $\epsilon _1, \ldots, \epsilon _k\in\{ \pm 1\}$ and a neighbourhood $V$ of $x$ such that $h = h_1^{\epsilon _1}\circ\ldots\circ h_k^{\epsilon _k}$ on $V$. If $A$ is finite, $\Gamma (A)$ is said to be '''finitely generated'''. | Any set $A$ of homeomorphisms bewteen open sets (with domains covering a space $X$) generates ma pseudogroup $\Gamma (A)$ which is the smallest pseudogroup containing $A$; precisely a homeomorphism $h:D_h\to R_h$ belongs to $\Gamma (A)$ if and only if for any point $x\in D_h$ there exist elements $h_1, \ldots, h_k\in A$, exponents $\epsilon _1, \ldots, \epsilon _k\in\{ \pm 1\}$ and a neighbourhood $V$ of $x$ such that $h = h_1^{\epsilon _1}\circ\ldots\circ h_k^{\epsilon _k}$ on $V$. If $A$ is finite, $\Gamma (A)$ is said to be '''finitely generated'''. | ||
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</wikitex> | </wikitex> | ||
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For most of results listed here we refer to \cite{Hector&Hirsch1981}. | For most of results listed here we refer to \cite{Hector&Hirsch1981}. | ||
− | Let $G$ be a finitely generated group and $G_1$ a finite | + | Let $G$ be a finitely generated group and $G_1$ a finite \it{symmetric} |
(i.e. such that $e\in G_1$ and $G_1^{-1} = \{ g^{-1};\, g\in G_1\}\subset G_1$) | (i.e. such that $e\in G_1$ and $G_1^{-1} = \{ g^{-1};\, g\in G_1\}\subset G_1$) | ||
set generating it. For any $n\in\Bbb N$ let | set generating it. For any $n\in\Bbb N$ let | ||
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{{endthm}} | {{endthm}} | ||
− | {{beginthm|Theorem}} Any finitely generated nilpotent group is of polynomial type of growth. Moreover, if a finitely generated group contains a nilpotent group of finite index, then it has a polynomial type of growth \cite{Wolf1968}. Conversely, any finitely group of polynomial type of growth contains a nilpotent subgroup of finite index \cite{Gromov1981}. | + | {{beginthm|Theorem}} Any finitely generated nilpotent group is of polynomial type of growth. Moreover, if a finitely generated group contains a nilpotent group of finite index, then it has a polynomial type of growth \cite{Wolf1968}. Conversely, any finitely generated group of polynomial type of growth contains a nilpotent subgroup of finite index \cite{Gromov1981}. |
{{endthm}} | {{endthm}} | ||
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{{beginthm|Definition}} $gr (M) = [t_M]$ is called the '''growth type''' of a Riemannian manifold $M$. | {{beginthm|Definition}} $gr (M) = [t_M]$ is called the '''growth type''' of a Riemannian manifold $M$. | ||
{{endthm}} | {{endthm}} | ||
+ | |||
+ | {{beginthm|Proposition|(\cite{Milnor1968a}, \cite{Švarc1955})}} If $M$ is compact, then the fundamental group $\pi_1 (M)$ and the universal covering $\tilde M$ of $M$ have the same growth type. If $M$ is a compact Riemannian manifold of negative sectional | ||
+ | curvature, then both $\tilde M$ and $\pi_1(M)$ have exponential growth. | ||
+ | {{endthm}} | ||
+ | |||
+ | Fundamental groups of compact manifolds of negative sectional curvature are hyperbolic in the sense of \cite{Gromov1987}. The last statement of the Proposition above can be generalized as follows, | ||
+ | |||
+ | {{beginthm|Proposition}} Any non-elementary hyperbolic group has exponential growth. | ||
+ | {{endthm}} | ||
</wikitex> | </wikitex> | ||
Latest revision as of 17:40, 10 December 2010
The user responsible for this page is Pawel Walczak. No other user may edit this page at present. |
This page has not been refereed. The information given here might be incomplete or provisional. |
Contents |
1 Introduction
Foliations of Riemannian manifolds can be considered as multidimensional dynamical systems: Riemannian structure (more precisely, Riemannian structure along the leaves) plays the role of time, the leaves play the role of orbits [Walczak2004]. The dynamics of a foliation can be described in terms of its holonomy (see Foliations#Holonomy) pseudogroup. As far as we know, the notion of a pseudogroup appeared for the first time in [Veblen&Whitehead1932] in the context of geometric objects studied in differential geometry. Pseudogroups have been brought into the foliation theory by Haefliger [Haefliger1962a]. The growth types of leaves (equivalent to the growth types of the corresponding orbits of holonomy pseudogroups) as well as the expansion growth of a foliation (or, its holonomy pseudogroup) describe some aspects of the foliation dynamics. Corresponding to the expansion growth is the notion of geometric entropy of a foliation [Ghys&Langevin&Walczak1988]. Vanishing entropy can be seen as a dynamical condition which occurs to have strong geometric and topological consequences (see Dynamics of foliations#Results on entropy) below.
2 Pseudogroups
The notion of a pseudogroup generalizes that of a group of transformations. Given a space , any group of transformations of consists of maps defined globally on , mapping bijectively onto itself and such that the composition of any two of them as well as the inverse of any of them belongs to the group. The same holds for a pseudogroup with this difference that the maps are not defined globally but on open subsets, so the domain of the composition is usually smaller than those of the maps being composed.
To make the above precise, let us take a topological space and denote by Homeo the family of all homeomorphisms between open subsets of . If Homeo, then is its domain and .
Definition 2.1. A subfamily of Homeo is said to be a pseudogroup if it is closed under composition, inversion, restriction to open subdomains and unions. More precisely, should satisfy the following conditions:
(i) whenever and ,
(ii) whenever ,
(iii) whenever and is open,
(iv) if Homeo, is an open cover of and for any , then .
Moreover, we shall always assume that
(v] id (or, equivalently, ).
As examples, one may have (1) all the homeomorphisms between open sets of a topological space, (2) all the diffeomorhisms of a given class (say, , , between open sets of a manifold, (3) all the restrictions to open domains of elements of a given group of homeomorphisms of a given topological space. (In the last case, we say that the pseudogroup is generated by .) Any set of homeomorphisms bewteen open sets (with domains covering a space ) generates ma pseudogroup which is the smallest pseudogroup containing ; precisely a homeomorphism belongs to if and only if for any point there exist elements , exponents and a neighbourhood of such that on . If is finite, is said to be finitely generated.
3 Holonomy pesudogroups
Definition 3.1. A foliated atlas on a foliated manifold is said to be nice (also, nice is the covering of by the domains of the charts ) if
(i) the covering is locally finite,
(ii) for any , is an open cube,
(iii) if and , and , then there exists a foliated chart chart and such that is an open cube, contains the closure of and .
Since manifolds are supposed to be paracompact here, they are separable and hence nice coverings are denumerable. Nice coverings on compact manifolds are finite. On arbitrary foliated manfiolds, nice coverings do exist.
Given a nice covering of a foliated manifold . For any , let be the space of the plaques (i.e., connected components of intersections . being a leaf of ) of contained in . Equip with the quotient topology: two points of are equivalent iff they belong to the same plaque. is homeomorphic (-diffeomorphic when is -differentiable and ) to an open cube () via the map , where is a foliated chart on . The disjoint union
is called a complete transversal for . Transversality refers to the fact that, if is differentiable of class , , each of the spaces can be mapped homeomorphically onto a -submanifold transverse to : if and is the leaf of passing through , then
Completeness of means that every leaf of intersects at least one of the submanifolds .
Definition 3.2. Given a nice covering of a foliated manifold and two sets and such that the holonomy map , being the open subset of which consists of all the plaques of for which , is defined in the following way:
All the maps () generate a pseudogroup on . is called the holonomy pseudogroup of .
This means that any element of assigns to a plaque the end plaque of a chain (that is a sequence of plaques such that any two consequtive plaques intersect) which oroginates at .
Certainly, holonomy pseudogroups of arbitrary compact foliated manifolds are finitely generated. For example, gluing together two 2-dimensional Reeb foliations (see Foliations#Reeb Foliations) on one gets a foliation of the 3-dimensional sphere for which any arc intersecting the unique toral leaf is a complete transversal; can be identified with a segment (), the point of intersection with the number , while the holonomy group with the one on our segment generated by two maps such that , , snd contract their domains towards .
4 Growth
Let us begin with two non-decreasing sequences and of non-negative numbers. We shall say that "grows slower" that () whenever there exist positive constants and such that the inequalities
hold for all . We say that types of growth of our sequences and are the same whenever
Let now be the set of non-negative increasing functions defined on :
and the set of increasing sequences with entries in :
Elements of can be identified with sequences , so can be considered as a subset of .
A preorder defined by the condition:
(*) if and only if there exists such that for all the inequalities
induces an equivalence relation in :
In particular, if and , then
and
for all and some . (Here, is the largest integer which does not exceed , .)
Definition 4.1. Elements of the quotient are called growth types. The growth type of (resp., of ) is denoted by (resp., by ). Also, we let . is the set of growth types of monotone functions (in the sense of [Hector&Hirsch1981]). The preorder induces a partial order (denoted again by ) in .
Example 4.2. and all the growth types listed above are different. The growth type of any polynomial of degree is equal to and is called polynomial (of degree ). for any and this growth type is called exponential.
4.1 Growth in groups
For most of results listed here we refer to [Hector&Hirsch1981].
Let be a finitely generated group and a finite symmetric (i.e. such that and ) set generating it. For any let
and
The type of growth of does not depend on , so we may write the following.
Definition 4.3. The growth type of is defined as for any finite symmetric generating set . If acts on a space and , then the growth type of at is defined in a similar way: , where
for any fixed finite symmetric generating set .
Example 4.4. A finite group has the growth type while the abelian group has the polynomial growth of degree . Any free (non-abelian) group has the exponential growth .
Proposition 4.5. For any finitely generated group and any normal subgroup of we have
Proposition 4.6. Any finitely generated abelian group of rank has the growth type .
Theorem 4.7. Any finitely generated nilpotent group is of polynomial type of growth. Moreover, if a finitely generated group contains a nilpotent group of finite index, then it has a polynomial type of growth [Wolf1968]. Conversely, any finitely generated group of polynomial type of growth contains a nilpotent subgroup of finite index [Gromov1981].
Natural examples of finitely generated groups are provided by fundamental groups of compact manifolds. Below we show how the growth type of the fundamental group of a compact Riemannian manifold is related to its geometry. To this end let us consider a complete Riemannian manifold , fix a point and let
where is the ball on of radius and centre while is the measure on induced by the Riemannian structure. If is another point of , then , where and is the distance function on . Therefore, the following definition is correct.
Definition 4.8. is called the growth type of a Riemannian manifold .
Proposition 4.9 ([Milnor1968a], [Švarc1955]). If is compact, then the fundamental group and the universal covering of have the same growth type. If is a compact Riemannian manifold of negative sectional curvature, then both and have exponential growth.
Fundamental groups of compact manifolds of negative sectional curvature are hyperbolic in the sense of [Gromov1987]. The last statement of the Proposition above can be generalized as follows,
Proposition 4.10. Any non-elementary hyperbolic group has exponential growth.
4.2 Orbit growth in pseudogroups
4.3 Expansion growth
5 Geometric entropy
6 Invariant measures
7 Results on entropy
8 References
- [Ghys&Langevin&Walczak1988] É. Ghys, R. Langevin and P. Walczak, Entropie géométrique des feuilletages, Acta Math. 160 (1988), no.1-2, 105–142. MR926526 (89a:57034) Zbl 0666.57021
- [Gromov1981] M. Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math. (1981), no.53, 53–73. MR623534 (83b:53041) Zbl 0474.20018
- [Gromov1987] M. Gromov, Hyperbolic groups, Essays in group theory,Math. Sci. Res. Inst. Publ., 8, Springer, New York, (1987) 75–263. MR0919829 (89e:20070) Zbl 0634.20015
- [Haefliger1962a] A. Haefliger, Variétés feuilletées, Ann. Scuola Norm. Sup. Pisa (3) 16 (1962), 367–397. MR0189060 (32 #6487) Zbl 0196.25005
- [Hector&Hirsch1981] G. Hector and U. Hirsch, Introduction to the geometry of foliations. Part A, Friedr. Vieweg \& Sohn, Braunschweig, 1981. MR639738 (83d:57019) Zbl 0628.57001
- [Milnor1968a] J. Milnor, A note on curvature and fundamental group, J. Differential Geometry 2 (1968), 1–7. MR0232311 (38 #636) Zbl 0162.25401
- [Veblen&Whitehead1932] O. Veblen and J. Whitehead, The foundations of differential geometry, Cambridge Tracts in Math. a. Math. Phys. 29) London: Cambridge Univ. Press. IX, 97 S., 1932. Zbl 0005.21801
- [Walczak2004] P. Walczak, Dynamics of foliations, groups and pseudogroups, Birkhäuser Verlag, 2004. MR2056374 (2005d:57042) Zbl 1084.37022
- [Wolf1968] J. A. Wolf, Growth of finitely generated solvable groups and curvature of Riemanniann manifolds, J. Differential Geometry 2 (1968), 421–446. MR0248688 (40 #1939) Zbl 0207.51803
- [Švarc1955] A. S. Švarc, A volume invariant of coverings, Dokl. Akad. Nauk SSSR (N.S.) 105 (1955), 32–34. MR0075634 (17,781d)
To make the above precise, let us take a topological space and denote by Homeo the family of all homeomorphisms between open subsets of . If Homeo, then is its domain and .
Definition 2.1. A subfamily of Homeo is said to be a pseudogroup if it is closed under composition, inversion, restriction to open subdomains and unions. More precisely, should satisfy the following conditions:
(i) whenever and ,
(ii) whenever ,
(iii) whenever and is open,
(iv) if Homeo, is an open cover of and for any , then .
Moreover, we shall always assume that
(v] id (or, equivalently, ).
As examples, one may have (1) all the homeomorphisms between open sets of a topological space, (2) all the diffeomorhisms of a given class (say, , , between open sets of a manifold, (3) all the restrictions to open domains of elements of a given group of homeomorphisms of a given topological space. (In the last case, we say that the pseudogroup is generated by .) Any set of homeomorphisms bewteen open sets (with domains covering a space ) generates ma pseudogroup which is the smallest pseudogroup containing ; precisely a homeomorphism belongs to if and only if for any point there exist elements , exponents and a neighbourhood of such that on . If is finite, is said to be finitely generated.
3 Holonomy pesudogroups
Definition 3.1. A foliated atlas on a foliated manifold is said to be nice (also, nice is the covering of by the domains of the charts ) if
(i) the covering is locally finite,
(ii) for any , is an open cube,
(iii) if and , and , then there exists a foliated chart chart and such that is an open cube, contains the closure of and .
Since manifolds are supposed to be paracompact here, they are separable and hence nice coverings are denumerable. Nice coverings on compact manifolds are finite. On arbitrary foliated manfiolds, nice coverings do exist.
Given a nice covering of a foliated manifold . For any , let be the space of the plaques (i.e., connected components of intersections . being a leaf of ) of contained in . Equip with the quotient topology: two points of are equivalent iff they belong to the same plaque. is homeomorphic (-diffeomorphic when is -differentiable and ) to an open cube () via the map , where is a foliated chart on . The disjoint union
is called a complete transversal for . Transversality refers to the fact that, if is differentiable of class , , each of the spaces can be mapped homeomorphically onto a -submanifold transverse to : if and is the leaf of passing through , then
Completeness of means that every leaf of intersects at least one of the submanifolds .
Definition 3.2. Given a nice covering of a foliated manifold and two sets and such that the holonomy map , being the open subset of which consists of all the plaques of for which , is defined in the following way:
All the maps () generate a pseudogroup on . is called the holonomy pseudogroup of .
This means that any element of assigns to a plaque the end plaque of a chain (that is a sequence of plaques such that any two consequtive plaques intersect) which oroginates at .
Certainly, holonomy pseudogroups of arbitrary compact foliated manifolds are finitely generated. For example, gluing together two 2-dimensional Reeb foliations (see Foliations#Reeb Foliations) on one gets a foliation of the 3-dimensional sphere for which any arc intersecting the unique toral leaf is a complete transversal; can be identified with a segment (), the point of intersection with the number , while the holonomy group with the one on our segment generated by two maps such that , , snd contract their domains towards .
4 Growth
Let us begin with two non-decreasing sequences and of non-negative numbers. We shall say that "grows slower" that () whenever there exist positive constants and such that the inequalities
hold for all . We say that types of growth of our sequences and are the same whenever
Let now be the set of non-negative increasing functions defined on :
and the set of increasing sequences with entries in :
Elements of can be identified with sequences , so can be considered as a subset of .
A preorder defined by the condition:
(*) if and only if there exists such that for all the inequalities
induces an equivalence relation in :
In particular, if and , then
and
for all and some . (Here, is the largest integer which does not exceed , .)
Definition 4.1. Elements of the quotient are called growth types. The growth type of (resp., of ) is denoted by (resp., by ). Also, we let . is the set of growth types of monotone functions (in the sense of [Hector&Hirsch1981]). The preorder induces a partial order (denoted again by ) in .
Example 4.2. and all the growth types listed above are different. The growth type of any polynomial of degree is equal to and is called polynomial (of degree ). for any and this growth type is called exponential.
4.1 Growth in groups
For most of results listed here we refer to [Hector&Hirsch1981].
Let be a finitely generated group and a finite symmetric (i.e. such that and ) set generating it. For any let
and
The type of growth of does not depend on , so we may write the following.
Definition 4.3. The growth type of is defined as for any finite symmetric generating set . If acts on a space and , then the growth type of at is defined in a similar way: , where
for any fixed finite symmetric generating set .
Example 4.4. A finite group has the growth type while the abelian group has the polynomial growth of degree . Any free (non-abelian) group has the exponential growth .
Proposition 4.5. For any finitely generated group and any normal subgroup of we have
Proposition 4.6. Any finitely generated abelian group of rank has the growth type .
Theorem 4.7. Any finitely generated nilpotent group is of polynomial type of growth. Moreover, if a finitely generated group contains a nilpotent group of finite index, then it has a polynomial type of growth [Wolf1968]. Conversely, any finitely generated group of polynomial type of growth contains a nilpotent subgroup of finite index [Gromov1981].
Natural examples of finitely generated groups are provided by fundamental groups of compact manifolds. Below we show how the growth type of the fundamental group of a compact Riemannian manifold is related to its geometry. To this end let us consider a complete Riemannian manifold , fix a point and let
where is the ball on of radius and centre while is the measure on induced by the Riemannian structure. If is another point of , then , where and is the distance function on . Therefore, the following definition is correct.
Definition 4.8. is called the growth type of a Riemannian manifold .
Proposition 4.9 ([Milnor1968a], [Švarc1955]). If is compact, then the fundamental group and the universal covering of have the same growth type. If is a compact Riemannian manifold of negative sectional curvature, then both and have exponential growth.
Fundamental groups of compact manifolds of negative sectional curvature are hyperbolic in the sense of [Gromov1987]. The last statement of the Proposition above can be generalized as follows,
Proposition 4.10. Any non-elementary hyperbolic group has exponential growth.
4.2 Orbit growth in pseudogroups
4.3 Expansion growth
5 Geometric entropy
6 Invariant measures
7 Results on entropy
8 References
- [Ghys&Langevin&Walczak1988] É. Ghys, R. Langevin and P. Walczak, Entropie géométrique des feuilletages, Acta Math. 160 (1988), no.1-2, 105–142. MR926526 (89a:57034) Zbl 0666.57021
- [Gromov1981] M. Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math. (1981), no.53, 53–73. MR623534 (83b:53041) Zbl 0474.20018
- [Gromov1987] M. Gromov, Hyperbolic groups, Essays in group theory,Math. Sci. Res. Inst. Publ., 8, Springer, New York, (1987) 75–263. MR0919829 (89e:20070) Zbl 0634.20015
- [Haefliger1962a] A. Haefliger, Variétés feuilletées, Ann. Scuola Norm. Sup. Pisa (3) 16 (1962), 367–397. MR0189060 (32 #6487) Zbl 0196.25005
- [Hector&Hirsch1981] G. Hector and U. Hirsch, Introduction to the geometry of foliations. Part A, Friedr. Vieweg \& Sohn, Braunschweig, 1981. MR639738 (83d:57019) Zbl 0628.57001
- [Milnor1968a] J. Milnor, A note on curvature and fundamental group, J. Differential Geometry 2 (1968), 1–7. MR0232311 (38 #636) Zbl 0162.25401
- [Veblen&Whitehead1932] O. Veblen and J. Whitehead, The foundations of differential geometry, Cambridge Tracts in Math. a. Math. Phys. 29) London: Cambridge Univ. Press. IX, 97 S., 1932. Zbl 0005.21801
- [Walczak2004] P. Walczak, Dynamics of foliations, groups and pseudogroups, Birkhäuser Verlag, 2004. MR2056374 (2005d:57042) Zbl 1084.37022
- [Wolf1968] J. A. Wolf, Growth of finitely generated solvable groups and curvature of Riemanniann manifolds, J. Differential Geometry 2 (1968), 421–446. MR0248688 (40 #1939) Zbl 0207.51803
- [Švarc1955] A. S. Švarc, A volume invariant of coverings, Dokl. Akad. Nauk SSSR (N.S.) 105 (1955), 32–34. MR0075634 (17,781d)
To make the above precise, let us take a topological space and denote by Homeo the family of all homeomorphisms between open subsets of . If Homeo, then is its domain and .
Definition 2.1. A subfamily of Homeo is said to be a pseudogroup if it is closed under composition, inversion, restriction to open subdomains and unions. More precisely, should satisfy the following conditions:
(i) whenever and ,
(ii) whenever ,
(iii) whenever and is open,
(iv) if Homeo, is an open cover of and for any , then .
Moreover, we shall always assume that
(v] id (or, equivalently, ).
As examples, one may have (1) all the homeomorphisms between open sets of a topological space, (2) all the diffeomorhisms of a given class (say, , , between open sets of a manifold, (3) all the restrictions to open domains of elements of a given group of homeomorphisms of a given topological space. (In the last case, we say that the pseudogroup is generated by .) Any set of homeomorphisms bewteen open sets (with domains covering a space ) generates ma pseudogroup which is the smallest pseudogroup containing ; precisely a homeomorphism belongs to if and only if for any point there exist elements , exponents and a neighbourhood of such that on . If is finite, is said to be finitely generated.
3 Holonomy pesudogroups
Definition 3.1. A foliated atlas on a foliated manifold is said to be nice (also, nice is the covering of by the domains of the charts ) if
(i) the covering is locally finite,
(ii) for any , is an open cube,
(iii) if and , and , then there exists a foliated chart chart and such that is an open cube, contains the closure of and .
Since manifolds are supposed to be paracompact here, they are separable and hence nice coverings are denumerable. Nice coverings on compact manifolds are finite. On arbitrary foliated manfiolds, nice coverings do exist.
Given a nice covering of a foliated manifold . For any , let be the space of the plaques (i.e., connected components of intersections . being a leaf of ) of contained in . Equip with the quotient topology: two points of are equivalent iff they belong to the same plaque. is homeomorphic (-diffeomorphic when is -differentiable and ) to an open cube () via the map , where is a foliated chart on . The disjoint union
is called a complete transversal for . Transversality refers to the fact that, if is differentiable of class , , each of the spaces can be mapped homeomorphically onto a -submanifold transverse to : if and is the leaf of passing through , then
Completeness of means that every leaf of intersects at least one of the submanifolds .
Definition 3.2. Given a nice covering of a foliated manifold and two sets and such that the holonomy map , being the open subset of which consists of all the plaques of for which , is defined in the following way:
All the maps () generate a pseudogroup on . is called the holonomy pseudogroup of .
This means that any element of assigns to a plaque the end plaque of a chain (that is a sequence of plaques such that any two consequtive plaques intersect) which oroginates at .
Certainly, holonomy pseudogroups of arbitrary compact foliated manifolds are finitely generated. For example, gluing together two 2-dimensional Reeb foliations (see Foliations#Reeb Foliations) on one gets a foliation of the 3-dimensional sphere for which any arc intersecting the unique toral leaf is a complete transversal; can be identified with a segment (), the point of intersection with the number , while the holonomy group with the one on our segment generated by two maps such that , , snd contract their domains towards .
4 Growth
Let us begin with two non-decreasing sequences and of non-negative numbers. We shall say that "grows slower" that () whenever there exist positive constants and such that the inequalities
hold for all . We say that types of growth of our sequences and are the same whenever
Let now be the set of non-negative increasing functions defined on :
and the set of increasing sequences with entries in :
Elements of can be identified with sequences , so can be considered as a subset of .
A preorder defined by the condition:
(*) if and only if there exists such that for all the inequalities
induces an equivalence relation in :
In particular, if and , then
and
for all and some . (Here, is the largest integer which does not exceed , .)
Definition 4.1. Elements of the quotient are called growth types. The growth type of (resp., of ) is denoted by (resp., by ). Also, we let . is the set of growth types of monotone functions (in the sense of [Hector&Hirsch1981]). The preorder induces a partial order (denoted again by ) in .
Example 4.2. and all the growth types listed above are different. The growth type of any polynomial of degree is equal to and is called polynomial (of degree ). for any and this growth type is called exponential.
4.1 Growth in groups
For most of results listed here we refer to [Hector&Hirsch1981].
Let be a finitely generated group and a finite symmetric (i.e. such that and ) set generating it. For any let
and
The type of growth of does not depend on , so we may write the following.
Definition 4.3. The growth type of is defined as for any finite symmetric generating set . If acts on a space and , then the growth type of at is defined in a similar way: , where
for any fixed finite symmetric generating set .
Example 4.4. A finite group has the growth type while the abelian group has the polynomial growth of degree . Any free (non-abelian) group has the exponential growth .
Proposition 4.5. For any finitely generated group and any normal subgroup of we have
Proposition 4.6. Any finitely generated abelian group of rank has the growth type .
Theorem 4.7. Any finitely generated nilpotent group is of polynomial type of growth. Moreover, if a finitely generated group contains a nilpotent group of finite index, then it has a polynomial type of growth [Wolf1968]. Conversely, any finitely generated group of polynomial type of growth contains a nilpotent subgroup of finite index [Gromov1981].
Natural examples of finitely generated groups are provided by fundamental groups of compact manifolds. Below we show how the growth type of the fundamental group of a compact Riemannian manifold is related to its geometry. To this end let us consider a complete Riemannian manifold , fix a point and let
where is the ball on of radius and centre while is the measure on induced by the Riemannian structure. If is another point of , then , where and is the distance function on . Therefore, the following definition is correct.
Definition 4.8. is called the growth type of a Riemannian manifold .
Proposition 4.9 ([Milnor1968a], [Švarc1955]). If is compact, then the fundamental group and the universal covering of have the same growth type. If is a compact Riemannian manifold of negative sectional curvature, then both and have exponential growth.
Fundamental groups of compact manifolds of negative sectional curvature are hyperbolic in the sense of [Gromov1987]. The last statement of the Proposition above can be generalized as follows,
Proposition 4.10. Any non-elementary hyperbolic group has exponential growth.
4.2 Orbit growth in pseudogroups
4.3 Expansion growth
5 Geometric entropy
6 Invariant measures
7 Results on entropy
8 References
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