the large scale geometry - kobe universitygentop.h.kobe-u.ac.jp/slides/virk.pdf · the large scale...
TRANSCRIPT
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
The large scale geometry
Ziga Virk
University of Ljubljana, Slovenia
General Topology Symposium 2012, Kobe, Japan
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
The idea: to study the metric spaces from a large perspective.Compact sets ⇡ points.
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
The idea: to study the metric spaces from a large perspective.Compact sets ⇡ points.
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
The idea: to study the metric spaces from a large perspective.Compact sets ⇡ points.
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
The idea: to study the metric spaces from a large perspective.Compact sets ⇡ points.
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
The idea: to study the metric spaces from a large perspective.Compact sets ⇡ points.
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
The idea: to study the metric spaces from a large perspective.Compact sets ⇡ points.
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
The idea: to study the metric spaces from a large perspective.Compact sets ⇡ points.
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
The idea: to study the metric spaces from a large perspective.Compact sets ⇡ points.
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
The idea: to study the metric spaces from a large perspective.Compact sets ⇡ points.
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
The idea: to study the metric spaces from a large perspective.Compact sets ⇡ points.
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
f : X ! Y a map between metric spaces (no continuityrequired).
• f is proper if f �1(A) is bounded for every bounded subsetA ⇢ Y ;
• f is bornologous (or large scale uniform, or ls-uniform)8R < 1, 9S < 1 :
d(x , y) < R ) d(f (x), f (y)) < S
• f is coarse if it is proper and bornologous;
• f is close (or ls-equivalent) to g : Z ! Y if9R > 0 : d(f (x), g(x)) < R , 8x 2 X . [notation: f ⇠
ls
g ]
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
f : X ! Y a map between metric spaces (no continuityrequired).
• f is proper if f �1(A) is bounded for every bounded subsetA ⇢ Y ;
• f is bornologous (or large scale uniform, or ls-uniform)8R < 1, 9S < 1 :
d(x , y) < R ) d(f (x), f (y)) < S
• f is coarse if it is proper and bornologous;
• f is close (or ls-equivalent) to g : Z ! Y if9R > 0 : d(f (x), g(x)) < R , 8x 2 X . [notation: f ⇠
ls
g ]
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
Coarse category
• Objects: metric spaces;
• Morphisms: [equivalence classes of] coarse maps;
• Spaces X and Y are coarsely equivalent [X ⇠ls
Y ], ifthere exist coarse maps f : X ! Y and g : Y ! X so thatfg ⇠
ls
1Y
and gf ⇠ls
1X
;
• A special case of a bornologous map: ls-Lipschitz map (or(c ,A)�Lipschitz map). They possess a linear bound onthe size of the image:
d(f (x), f (y)) cd(x , y) + A, 8x , y 2 X
• (c ,A)�bilipschitz maps:
c�1d(x , y)�A d(f (x), f (y)) cd(x , y)+A, 8x , y 2 X
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
Coarse category
• Objects: metric spaces;
• Morphisms: [equivalence classes of] coarse maps;
• Spaces X and Y are coarsely equivalent [X ⇠ls
Y ], ifthere exist coarse maps f : X ! Y and g : Y ! X so thatfg ⇠
ls
1Y
and gf ⇠ls
1X
;
• A special case of a bornologous map: ls-Lipschitz map (or(c ,A)�Lipschitz map). They possess a linear bound onthe size of the image:
d(f (x), f (y)) cd(x , y) + A, 8x , y 2 X
• (c ,A)�bilipschitz maps:
c�1d(x , y)�A d(f (x), f (y)) cd(x , y)+A, 8x , y 2 X
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
Coarse category
• Objects: metric spaces;
• Morphisms: [equivalence classes of] coarse maps;
• Spaces X and Y are coarsely equivalent [X ⇠ls
Y ], ifthere exist coarse maps f : X ! Y and g : Y ! X so thatfg ⇠
ls
1Y
and gf ⇠ls
1X
;
• A special case of a bornologous map: ls-Lipschitz map (or(c ,A)�Lipschitz map). They possess a linear bound onthe size of the image:
d(f (x), f (y)) cd(x , y) + A, 8x , y 2 X
• (c ,A)�bilipschitz maps:
c�1d(x , y)�A d(f (x), f (y)) cd(x , y)+A, 8x , y 2 X
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
Coarse category
• Objects: metric spaces;
• Morphisms: [equivalence classes of] coarse maps;
• Spaces X and Y are coarsely equivalent [X ⇠ls
Y ], ifthere exist coarse maps f : X ! Y and g : Y ! X so thatfg ⇠
ls
1Y
and gf ⇠ls
1X
;
• A special case of a bornologous map: ls-Lipschitz map (or(c ,A)�Lipschitz map). They possess a linear bound onthe size of the image:
d(f (x), f (y)) cd(x , y) + A, 8x , y 2 X
• (c ,A)�bilipschitz maps:
c�1d(x , y)�A d(f (x), f (y)) cd(x , y)+A, 8x , y 2 X
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
• ls-Lipschitz correspond with coarse maps in the usualsetting of (coarsely) geodesic spaces;
• Rn ⇠ls
Zn;
• X ⇠ls
⇤ i↵ X is bounded;
• for every metric space X there exists a discrete Y :X ⇠
ls
Y .
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
• ls-Lipschitz correspond with coarse maps in the usualsetting of (coarsely) geodesic spaces;
• Rn ⇠ls
Zn;
• X ⇠ls
⇤ i↵ X is bounded;
• for every metric space X there exists a discrete Y :X ⇠
ls
Y .
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
• ls-Lipschitz correspond with coarse maps in the usualsetting of (coarsely) geodesic spaces;
• Rn ⇠ls
Zn;
• X ⇠ls
⇤ i↵ X is bounded;
• for every metric space X there exists a discrete Y :X ⇠
ls
Y .
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
• ls-Lipschitz correspond with coarse maps in the usualsetting of (coarsely) geodesic spaces;
• Rn ⇠ls
Zn;
• X ⇠ls
⇤ i↵ X is bounded;
• for every metric space X there exists a discrete Y :X ⇠
ls
Y .
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
The group setting
G =< g1
, . . . , gn
| r1
, . . . > a finitely generated group.Cayley graph �
G
:
• vertices: elements of G ;
• edges: [u, v ] , 9i : u = vg±1
i
;
• loops in �G
are ”relations” on the generating set;
• �G
is geodesic;
• �G
depends on presentation, its coarse type does not.
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
The group setting
G =< g1
, . . . , gn
| r1
, . . . > a finitely generated group.Cayley graph �
G
:
• vertices: elements of G ;
• edges: [u, v ] , 9i : u = vg±1
i
;
• loops in �G
are ”relations” on the generating set;
• �G
is geodesic;
• �G
depends on presentation, its coarse type does not.
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
The group setting
G =< g1
, . . . , gn
| r1
, . . . > a finitely generated group.Cayley graph �
G
:
• vertices: elements of G ;
• edges: [u, v ] , 9i : u = vg±1
i
;
• loops in �G
are ”relations” on the generating set;
• �G
is geodesic;
• �G
depends on presentation, its coarse type does not.
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
e
a a b b
a2 ab ab a2 ab ab ba ba b2 ba ba b2
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
Coarse invariants of groups
Gromov initiated intense research.
• being finite / finitely presented;
• virtually Abelian;
• virtually Nilpotent;
• virtually free;
• amenable;
• hyperbolic;
• asymptotic dimension;
• ends of group;
• growth rate.
Group G hasvirtual property
P if there exists afinite indexsubgroup H Gwith property P.
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
Coarse invariants of groups
Gromov initiated intense research.
• being finite / finitely presented;
• virtually Abelian;
• virtually Nilpotent;
• virtually free;
• amenable;
• hyperbolic;
• asymptotic dimension;
• ends of group;
• growth rate.
Group G hasvirtual property
P if there exists afinite indexsubgroup H Gwith property P.
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
Application to Novikov conjecture by Yu:
• Novikov conjecture holds for groups of finite asymptoticdimension;
• Novikov conjecture holds for groups with property A.
Important aspect: coarse embeddings into Hilbert space.
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
Application to Novikov conjecture by Yu:
• Novikov conjecture holds for groups of finite asymptoticdimension;
• Novikov conjecture holds for groups with property A.
Important aspect: coarse embeddings into Hilbert space.
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
Application to Novikov conjecture by Yu:
• Novikov conjecture holds for groups of finite asymptoticdimension;
• Novikov conjecture holds for groups with property A.
Important aspect: coarse embeddings into Hilbert space.
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
Property A can be considered as a variant of the amenability.
A countable discrete group G is amenable if 8R , " > 0 exists afinitely supported f 2 `1(G ) such that:
1 ||f ||1
= 1;
2 ||gf � f ||1
< ", 8g 2 B(e,R).
A countable discrete group G has property A if there exists1 p < 1 so that 8R , " > 0 exists F : G ! `1(G ) with thefollowing properties:
1 ||Fx
||p
= 1, 8x 2 G ;
2 ||Fx
� Fy
||p
< ", 8x , y 2 G , d(x , y) < R ;
3 9S > 0 : supp Fx
⇢ B(x , S), 8x 2 G .
Free group on two elements has property A but is notamenable.
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
Property A can be considered as a variant of the amenability.
A countable discrete group G is amenable if 8R , " > 0 exists afinitely supported f 2 `1(G ) such that:
1 ||f ||1
= 1;
2 ||gf � f ||1
< ", 8g 2 B(e,R).
A countable discrete group G has property A if there exists1 p < 1 so that 8R , " > 0 exists F : G ! `1(G ) with thefollowing properties:
1 ||Fx
||p
= 1, 8x 2 G ;
2 ||Fx
� Fy
||p
< ", 8x , y 2 G , d(x , y) < R ;
3 9S > 0 : supp Fx
⇢ B(x , S), 8x 2 G .
Free group on two elements has property A but is notamenable.
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
Property A can be considered as a variant of the amenability.
A countable discrete group G is amenable if 8R , " > 0 exists afinitely supported f 2 `1(G ) such that:
1 ||f ||1
= 1;
2 ||gf � f ||1
< ", 8g 2 B(e,R).
A countable discrete group G has property A if there exists1 p < 1 so that 8R , " > 0 exists F : G ! `1(G ) with thefollowing properties:
1 ||Fx
||p
= 1, 8x 2 G ;
2 ||Fx
� Fy
||p
< ", 8x , y 2 G , d(x , y) < R ;
3 9S > 0 : supp Fx
⇢ B(x , S), 8x 2 G .
Free group on two elements has property A but is notamenable.
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
Property A can be considered as a variant of the amenability.
A countable discrete group G is amenable if 8R , " > 0 exists afinitely supported f 2 `1(G ) such that:
1 ||f ||1
= 1;
2 ||gf � f ||1
< ", 8g 2 B(e,R).
A countable discrete group G has property A if there exists1 p < 1 so that 8R , " > 0 exists F : G ! `1(G ) with thefollowing properties:
1 ||Fx
||p
= 1, 8x 2 G ;
2 ||Fx
� Fy
||p
< ", 8x , y 2 G , d(x , y) < R ;
3 9S > 0 : supp Fx
⇢ B(x , S), 8x 2 G .
Free group on two elements has property A but is notamenable.
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
Property A can be considered as a weakening of the finiteasymptotic dimension.
A discrete metric space of bounded geometry has property A,if 8R , " > 0, 9S > 0 and a family of finite nonempty subsets{A
x
}x2X of X ⇥ N so that:
1 if x , y 2 X , d(x , y) R then |Ax
�A
y
||A
x
\Ay
| < ";
2 if x 2 X and (y , n) 2 Ax
then d(x , y) S .
A discrete metric space of bounded geometry has asymptotic
dimension at most n if, additionally, the projection of Ax
ontoX has at most n + 1 elements, 8x 2 X , 8R , " > 0.
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
Property A can be considered as a weakening of the finiteasymptotic dimension.
A discrete metric space of bounded geometry has property A,if 8R , " > 0, 9S > 0 and a family of finite nonempty subsets{A
x
}x2X of X ⇥ N so that:
1 if x , y 2 X , d(x , y) R then |Ax
�A
y
||A
x
\Ay
| < ";
2 if x 2 X and (y , n) 2 Ax
then d(x , y) S .
A discrete metric space of bounded geometry has asymptotic
dimension at most n if, additionally, the projection of Ax
ontoX has at most n + 1 elements, 8x 2 X , 8R , " > 0.
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
The combinatorial approach
Idea: dualization of the shape theory. Representation of thecoarse structure by approximating complexes.Conventions:
• short map: Lipschitz map with constant 1;
• A(�) for a graph �: graph � with added edges. Forvertices u, v , d(u, v) = 2 the edge [u, v ] is added;
• A(�) of a graph �: A(�) is a complex of � so that � is asimplex of A(�) whenever [v ,w ] belongs to � for allv ,w 2 �;
• motivation: Rips graphs of a space.
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
The combinatorial approach
Idea: dualization of the shape theory. Representation of thecoarse structure by approximating complexes.Conventions:
• short map: Lipschitz map with constant 1;
• A(�) for a graph �: graph � with added edges. Forvertices u, v , d(u, v) = 2 the edge [u, v ] is added;
• A(�) of a graph �: A(�) is a complex of � so that � is asimplex of A(�) whenever [v ,w ] belongs to � for allv ,w 2 �;
• motivation: Rips graphs of a space.
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
Approximation by the system of
graphs
Objects: Coarse graphs.
A coarse graph is a direct sequence {V1
! V2
! . . .} ofgraphs V
n
and short maps in,m : V
n
! Vm
for all n m suchthat
1 in,n = id for all n � 1,
2 in,k = i
m,k � in,m for all n m k ,
3 for every n � 1 there is m > n so that in,m : A(V
n
) ! Vm
is short.
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
Approximation by the system of
graphs
Objects: Coarse graphs.
A coarse graph is a direct sequence {V1
! V2
! . . .} ofgraphs V
n
and short maps in,m : V
n
! Vm
for all n m suchthat
1 in,n = id for all n � 1,
2 in,k = i
m,k � in,m for all n m k ,
3 for every n � 1 there is m > n so that in,m : A(V
n
) ! Vm
is short.
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
Morphisms: equivalence classes of pre-morphisms.
Suppose V = {V1
i
1,2! V2
i
2,3! . . .} and W = {W1
j
1,2! W2
j
2,3! . . .}are two coarse graphs. A pre-morphism F : V ! W: afunction n
F
: N ! N, and short maps fk
: Vk
! Wn
F
(k)
so thatfor every k � 1 there is m � n
F
(k + 1) resulting injn
F
(k),m � fk
⇠ls
jn
F
(k+1),m � fk+1
� ik,k+1
.
W1
W2
· · · Wn
F
(1)
· · · Wn
F
(2)
· · · Wn
F
(3)
· · ·
V1
V2
V3
· · ·
j1,2 j
2,3 j j j j j
f1
i1,2
f2
i2,3
f1
i3,4
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
W1
W2
· · · Wn
F
(1)
· · · Wn
F
(2)
· · · Wn
F
(3)
· · ·
V1
V2
V3
· · ·
j1,2 j
2,3 j j j j j
f1
i1,2
f2
i2,3
f1
i3,4
Two pre-morphisms F ,G : V ! W are considered to beequivalent if for every k there is m � max{n
F
(k), nG
(k)} sothat j
n
F
(k),m � fk
⇠ls
jn
G
(k),m � gk
. The sets of equivalenceclasses of pre-morphisms form the set of morphisms from V toW.Coarse graphs V and W are ls-equivalent if there existpre-morphisms F : V ! W and G : W ! V such that G � F isequivalent to idV and F � G is equivalent to idW .
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
W1
W2
· · · Wn
F
(1)
· · · Wn
F
(2)
· · · Wn
F
(3)
· · ·
V1
V2
V3
· · ·
j1,2 j
2,3 j j j j j
f1
i1,2
f2
i2,3
f1
i3,4
Two pre-morphisms F ,G : V ! W are considered to beequivalent if for every k there is m � max{n
F
(k), nG
(k)} sothat j
n
F
(k),m � fk
⇠ls
jn
G
(k),m � gk
. The sets of equivalenceclasses of pre-morphisms form the set of morphisms from V toW.Coarse graphs V and W are ls-equivalent if there existpre-morphisms F : V ! W and G : W ! V such that G � F isequivalent to idV and F � G is equivalent to idW .
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
Associating a coarse graph to the metric space.
• Rips graphs RipsGr
for r ! 1:1 maps i are identity;2 [u, v ] is an edge in RipsG
r
i↵ d(u, v) r .
• Rips graph of an increasing sequence of uniformly boundedcovers U
n
for which Lebesgue number ! 1 :1 maps i are identity;2 [u, v ] is an edge in RipsGU
n
i↵ u and v lie in the sameelement of U
n
.
If V = {V1
! V2
! . . .} is a coarse graph of (X , dX
) andW = {W
1
! W2
! . . .} is a coarse graph of (Y , dY
), thenthere is a natural bijection between bornologous maps from Xto Y and morphisms from V to W.The description is su�cient for the formulation of the coarsetype but not for some other invariants.
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
Associating a coarse graph to the metric space.
• Rips graphs RipsGr
for r ! 1:1 maps i are identity;2 [u, v ] is an edge in RipsG
r
i↵ d(u, v) r .
• Rips graph of an increasing sequence of uniformly boundedcovers U
n
for which Lebesgue number ! 1 :1 maps i are identity;2 [u, v ] is an edge in RipsGU
n
i↵ u and v lie in the sameelement of U
n
.
If V = {V1
! V2
! . . .} is a coarse graph of (X , dX
) andW = {W
1
! W2
! . . .} is a coarse graph of (Y , dY
), thenthere is a natural bijection between bornologous maps from Xto Y and morphisms from V to W.The description is su�cient for the formulation of the coarsetype but not for some other invariants.
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
Associating a coarse graph to the metric space.
• Rips graphs RipsGr
for r ! 1:1 maps i are identity;2 [u, v ] is an edge in RipsG
r
i↵ d(u, v) r .
• Rips graph of an increasing sequence of uniformly boundedcovers U
n
for which Lebesgue number ! 1 :1 maps i are identity;2 [u, v ] is an edge in RipsGU
n
i↵ u and v lie in the sameelement of U
n
.
If V = {V1
! V2
! . . .} is a coarse graph of (X , dX
) andW = {W
1
! W2
! . . .} is a coarse graph of (Y , dY
), thenthere is a natural bijection between bornologous maps from Xto Y and morphisms from V to W.The description is su�cient for the formulation of the coarsetype but not for some other invariants.
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
Associating a coarse graph to the metric space.
• Rips graphs RipsGr
for r ! 1:1 maps i are identity;2 [u, v ] is an edge in RipsG
r
i↵ d(u, v) r .
• Rips graph of an increasing sequence of uniformly boundedcovers U
n
for which Lebesgue number ! 1 :1 maps i are identity;2 [u, v ] is an edge in RipsGU
n
i↵ u and v lie in the sameelement of U
n
.
If V = {V1
! V2
! . . .} is a coarse graph of (X , dX
) andW = {W
1
! W2
! . . .} is a coarse graph of (Y , dY
), thenthere is a natural bijection between bornologous maps from Xto Y and morphisms from V to W.The description is su�cient for the formulation of the coarsetype but not for some other invariants.
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
Approximation by the system of
complexes
Objects: Coarse complexes.
A coarse simplicial complex is a direct sequence{V
1
! V2
! . . .} of simplicial complexes Vn
and simplicialmaps i
n,m : Vn
! Vm
for all n m such that
1 in,n = id for all n � 1,
2 in,k = i
m,k � in,m for all n m k ,
3 for every n � 1 there is m > n so that in,m : A(V
n
) ! Vm
is short.
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
Approximation by the system of
complexes
Objects: Coarse complexes.
A coarse simplicial complex is a direct sequence{V
1
! V2
! . . .} of simplicial complexes Vn
and simplicialmaps i
n,m : Vn
! Vm
for all n m such that
1 in,n = id for all n � 1,
2 in,k = i
m,k � in,m for all n m k ,
3 for every n � 1 there is m > n so that in,m : A(V
n
) ! Vm
is short.
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
Morphisms: equivalence classes of pre-morphisms.
Suppose V = {V1
i
1,2! V2
i
2,3! . . .} and W = {W1
j
1,2! W2
j
2,3! . . .}are two coarse coarse simplicial complexes. A pre-morphism
F : V ! W: a function nF
: N ! N, and simplicial mapsfk
: Vk
! Wn
F
(k)
so that for every k � 1 there ism � n
F
(k + 1) resulting injn
F
(k),m � fk
⇠ls
jn
F
(k+1),m � fk+1
� ik,k+1
.
W1
W2
· · · Wn
F
(1)
· · · Wn
F
(2)
· · · Wn
F
(3)
· · ·
V1
V2
V3
· · ·
j1,2 j
2,3 j j j j j
f1
i1,2
f2
i2,3
f1
i3,4
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
Associating a coarse simplicial complex to the metric space.
• Rips complexes Ripsr
for r ! 1:1 maps i are identity;2 [u
0
, u1
, u2
, . . . , uk
] is a simplex in Ripsr
i↵Diam{u
0
, u1
, u2
, . . . , uk
} r .
• Rips complex of an increasing sequence of uniformlybounded covers U
n
for which Lebesgue number ! 1 :1 maps i are identity;2 [u
0
, u1
, u2
, . . . , uk
] is a simplex in RipsUn
i↵[u
0
, u1
, u2
, . . . , uk
] is a subset of some element of Un
.
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
Associating a coarse simplicial complex to the metric space.
• Rips complexes Ripsr
for r ! 1:1 maps i are identity;2 [u
0
, u1
, u2
, . . . , uk
] is a simplex in Ripsr
i↵Diam{u
0
, u1
, u2
, . . . , uk
} r .
• Rips complex of an increasing sequence of uniformlybounded covers U
n
for which Lebesgue number ! 1 :1 maps i are identity;2 [u
0
, u1
, u2
, . . . , uk
] is a simplex in RipsUn
i↵[u
0
, u1
, u2
, . . . , uk
] is a subset of some element of Un
.
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
Suppose Un
is a sequence of uniformly bounded covers of Xsuch that U
n�1
is a star refinement of Un
for each n � 1. Thenthe sequence N (U
1
) ! N (U2
) ! . . . of nerves of covers Un
forms a coarse simplicial complex if in,n+1
(U) contains the starst(U,U
n
) for each U 2 Un
. Any such coarse complex will bedenoted by Cech⇤(X ) and called a coarse
ˇ
Cech complex of X .
If V = {V1
! V2
! . . .} is a coarse simplicial complex of(X , d
X
) and W = {W1
! W2
! . . .} is a coarse simplicialcomplex of (Y , d
Y
), then there is a natural bijection betweenbornologous maps from X to Y and morphisms from V to W.
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
Suppose Un
is a sequence of uniformly bounded covers of Xsuch that U
n�1
is a star refinement of Un
for each n � 1. Thenthe sequence N (U
1
) ! N (U2
) ! . . . of nerves of covers Un
forms a coarse simplicial complex if in,n+1
(U) contains the starst(U,U
n
) for each U 2 Un
. Any such coarse complex will bedenoted by Cech⇤(X ) and called a coarse
ˇ
Cech complex of X .
If V = {V1
! V2
! . . .} is a coarse simplicial complex of(X , d
X
) and W = {W1
! W2
! . . .} is a coarse simplicialcomplex of (Y , d
Y
), then there is a natural bijection betweenbornologous maps from X to Y and morphisms from V to W.
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
Asymptotic dimension
Simplicial maps f , g : K ! L between simplicial complexes arecontiguous if for every simplex � of K , f (�) [ g(�) iscontained in some simplex of L.
Given a coarse simplicial complex K we say its asymptotic
dimension is at most n (notation: asdim(K) n) if for eachm there is k > m such that i
m,k factors contiguously throughan n-dimensional simplicial complex.
· · · Km
· · · Kk
· · ·
L(n)
im,k
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
Asymptotic dimension
Simplicial maps f , g : K ! L between simplicial complexes arecontiguous if for every simplex � of K , f (�) [ g(�) iscontained in some simplex of L.
Given a coarse simplicial complex K we say its asymptotic
dimension is at most n (notation: asdim(K) n) if for eachm there is k > m such that i
m,k factors contiguously throughan n-dimensional simplicial complex.
· · · Km
· · · Kk
· · ·
L(n)
im,k
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
Property A
A coarse simplicial complex K = {K1
! K2
! . . .} hasProperty A if for each k � 1 and each ✏ > 0 there is n > kand a function f : |K
k
|m
! |Kn
|m
such that f is contiguous toik,n : |Kk
| ! |Kn
| and the diameter of f (|�|) is at most ✏ foreach simplex � of K
k
.
· · · Kk
· · · Kn
· · ·
|Kk
|m
|Kn
|m
im,k
f
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
Coarse simple connectivity
A metric space X is coarsely k-connected if for each r thereexists R � r so that the mapping |Rips
r
(X )| ! |RipsR
(X )|induces a trivial map of ⇡
i
for 0 i k .
1 A finitely generated group is coarsely 1�connected i↵ it isfinitely presented;
2 [Fujiwara, White] Suppose X is a geodesic metric space.X is quasi-isometric to a simplicial tree if H
1
(X ) isuniformly generated and X is of asymptotic dimension 1;
3 An application of the previous item: finitely presentedgroups of asymptotic dimension 1 are virtually free;
4 If the fundamental group of a compact, connected, locallyconnected metric space is countable, then it is finitelypresented.
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
Coarse simple connectivity
A metric space X is coarsely k-connected if for each r thereexists R � r so that the mapping |Rips
r
(X )| ! |RipsR
(X )|induces a trivial map of ⇡
i
for 0 i k .
1 A finitely generated group is coarsely 1�connected i↵ it isfinitely presented;
2 [Fujiwara, White] Suppose X is a geodesic metric space.X is quasi-isometric to a simplicial tree if H
1
(X ) isuniformly generated and X is of asymptotic dimension 1;
3 An application of the previous item: finitely presentedgroups of asymptotic dimension 1 are virtually free;
4 If the fundamental group of a compact, connected, locallyconnected metric space is countable, then it is finitelypresented.
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
Coarse simple connectivity
A metric space X is coarsely k-connected if for each r thereexists R � r so that the mapping |Rips
r
(X )| ! |RipsR
(X )|induces a trivial map of ⇡
i
for 0 i k .
1 A finitely generated group is coarsely 1�connected i↵ it isfinitely presented;
2 [Fujiwara, White] Suppose X is a geodesic metric space.X is quasi-isometric to a simplicial tree if H
1
(X ) isuniformly generated and X is of asymptotic dimension 1;
3 An application of the previous item: finitely presentedgroups of asymptotic dimension 1 are virtually free;
4 If the fundamental group of a compact, connected, locallyconnected metric space is countable, then it is finitelypresented.
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
Coarse simple connectivity
A metric space X is coarsely k-connected if for each r thereexists R � r so that the mapping |Rips
r
(X )| ! |RipsR
(X )|induces a trivial map of ⇡
i
for 0 i k .
1 A finitely generated group is coarsely 1�connected i↵ it isfinitely presented;
2 [Fujiwara, White] Suppose X is a geodesic metric space.X is quasi-isometric to a simplicial tree if H
1
(X ) isuniformly generated and X is of asymptotic dimension 1;
3 An application of the previous item: finitely presentedgroups of asymptotic dimension 1 are virtually free;
4 If the fundamental group of a compact, connected, locallyconnected metric space is countable, then it is finitelypresented.
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
Coarse simple connectivity
A metric space X is coarsely k-connected if for each r thereexists R � r so that the mapping |Rips
r
(X )| ! |RipsR
(X )|induces a trivial map of ⇡
i
for 0 i k .
1 A finitely generated group is coarsely 1�connected i↵ it isfinitely presented;
2 [Fujiwara, White] Suppose X is a geodesic metric space.X is quasi-isometric to a simplicial tree if H
1
(X ) isuniformly generated and X is of asymptotic dimension 1;
3 An application of the previous item: finitely presentedgroups of asymptotic dimension 1 are virtually free;
4 If the fundamental group of a compact, connected, locallyconnected metric space is countable, then it is finitelypresented.
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
If the fundamental group of a compact, connected, locallyconnected metric space X is countable, then it is finitelypresented:
1 ⇡1
(X ) countable ) 9eX ;
2 X geodesic ) eX geodesic;
3 ⇡1
(X ) acts on eX ) [Svarc-Milnor Lemma] ⇡1
(X ) finitelygenerated and ⇡
1
(X ) 'ls
eX ;
4
eX coarsely simply connected ) ⇡1
(X ) coarsely simplyconnected;
5 ⇡1
(X ) finitely presented.
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
If the fundamental group of a compact, connected, locallyconnected metric space X is countable, then it is finitelypresented:
1 ⇡1
(X ) countable ) 9eX ;
2 X geodesic ) eX geodesic;
3 ⇡1
(X ) acts on eX ) [Svarc-Milnor Lemma] ⇡1
(X ) finitelygenerated and ⇡
1
(X ) 'ls
eX ;
4
eX coarsely simply connected ) ⇡1
(X ) coarsely simplyconnected;
5 ⇡1
(X ) finitely presented.
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
If the fundamental group of a compact, connected, locallyconnected metric space X is countable, then it is finitelypresented:
1 ⇡1
(X ) countable ) 9eX ;
2 X geodesic ) eX geodesic;
3 ⇡1
(X ) acts on eX ) [Svarc-Milnor Lemma] ⇡1
(X ) finitelygenerated and ⇡
1
(X ) 'ls
eX ;
4
eX coarsely simply connected ) ⇡1
(X ) coarsely simplyconnected;
5 ⇡1
(X ) finitely presented.
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
If the fundamental group of a compact, connected, locallyconnected metric space X is countable, then it is finitelypresented:
1 ⇡1
(X ) countable ) 9eX ;
2 X geodesic ) eX geodesic;
3 ⇡1
(X ) acts on eX ) [Svarc-Milnor Lemma] ⇡1
(X ) finitelygenerated and ⇡
1
(X ) 'ls
eX ;
4
eX coarsely simply connected ) ⇡1
(X ) coarsely simplyconnected;
5 ⇡1
(X ) finitely presented.
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
If the fundamental group of a compact, connected, locallyconnected metric space X is countable, then it is finitelypresented:
1 ⇡1
(X ) countable ) 9eX ;
2 X geodesic ) eX geodesic;
3 ⇡1
(X ) acts on eX ) [Svarc-Milnor Lemma] ⇡1
(X ) finitelygenerated and ⇡
1
(X ) 'ls
eX ;
4
eX coarsely simply connected ) ⇡1
(X ) coarsely simplyconnected;
5 ⇡1
(X ) finitely presented.
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
If the fundamental group of a compact, connected, locallyconnected metric space X is countable, then it is finitelypresented:
1 ⇡1
(X ) countable ) 9eX ;
2 X geodesic ) eX geodesic;
3 ⇡1
(X ) acts on eX ) [Svarc-Milnor Lemma] ⇡1
(X ) finitelygenerated and ⇡
1
(X ) 'ls
eX ;
4
eX coarsely simply connected ) ⇡1
(X ) coarsely simplyconnected;
5 ⇡1
(X ) finitely presented.
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
Partitions of unity
Given � > 0 and a simplicial complex K , f : X ! K is a��partition of unity if it is (�, �)�Lipschitz and{f �1(st(v))}
v2K (0)
is a uniformly bounded cover with Lebesguenumber at least ��1.
A metric space X is large scale paracompact i↵ for every� > 0 there exists a ��partition of unity.
A metric space of bounded geometry is large scale paracompacti↵ it has property A.
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
Partitions of unity
Given � > 0 and a simplicial complex K , f : X ! K is a��partition of unity if it is (�, �)�Lipschitz and{f �1(st(v))}
v2K (0)
is a uniformly bounded cover with Lebesguenumber at least ��1.
A metric space X is large scale paracompact i↵ for every� > 0 there exists a ��partition of unity.
A metric space of bounded geometry is large scale paracompacti↵ it has property A.
The large
scale
geometry
ˇ
Ziga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity
Partitions of unity
Given � > 0 and a simplicial complex K , f : X ! K is a��partition of unity if it is (�, �)�Lipschitz and{f �1(st(v))}
v2K (0)
is a uniformly bounded cover with Lebesguenumber at least ��1.
A metric space X is large scale paracompact i↵ for every� > 0 there exists a ��partition of unity.
A metric space of bounded geometry is large scale paracompacti↵ it has property A.