The large
scalegeometryŽiga Virk
The idea
The coarse
category
The group
setting

The large scale geometry

Motivation
The
combinatorial
approachŽiga Virk

Lipschitz
partitions ofunity

University of Ljubljana, Slovenia

General Topology Symposium 2012, Kobe, Japan

The large
scalegeometryŽiga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions ofunity

The idea: to study the metric spaces from a large perspective.
Compact sets ⇡ points.

The large
scalegeometryŽiga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions ofunity

The idea: to study the metric spaces from a large perspective.
Compact sets ⇡ points.

The large
scalegeometryŽiga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions ofunity

The idea: to study the metric spaces from a large perspective.
Compact sets ⇡ points.

The large
scalegeometryŽiga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions ofunity

The idea: to study the metric spaces from a large perspective.
Compact sets ⇡ points.

The large
scalegeometryŽiga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions ofunity

The idea: to study the metric spaces from a large perspective.
Compact sets ⇡ points.

The large
scalegeometryŽiga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions ofunity

The idea: to study the metric spaces from a large perspective.
Compact sets ⇡ points.

The large
scalegeometryŽiga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions ofunity

The idea: to study the metric spaces from a large perspective.
Compact sets ⇡ points.

The large
scalegeometryŽiga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions ofunity

The idea: to study the metric spaces from a large perspective.
Compact sets ⇡ points.

The large
scalegeometryŽiga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions ofunity

The idea: to study the metric spaces from a large perspective.
Compact sets ⇡ points.

The large
scalegeometryŽiga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions ofunity

The idea: to study the metric spaces from a large perspective.
Compact sets ⇡ points.

The large
scalegeometryŽiga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions ofunity

f : X ! Y a map between metric spaces (no continuity
required).
• f is proper if f 1 (A) is bounded for every bounded subset

A ⇢ 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 if

9R > 0 : d(f (x), g (x)) < R, 8x 2 X . [notation: f ⇠ls g ]The large
scalegeometryŽiga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions ofunity

f : X ! Y a map between metric spaces (no continuity
required).
• f is proper if f 1 (A) is bounded for every bounded subset

A ⇢ 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 if

9R > 0 : d(f (x), g (x)) < R, 8x 2 X . [notation: f ⇠ls g ]The large
scalegeometry

Coarse categoryŽiga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions ofunity• Objects: metric spaces;• Morphisms: [equivalence classes of] coarse maps;• Spaces X and Y are coarsely equivalent [X ⇠ls Y ], if

there 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 on
the size of the image:d(f (x), f (y ))  cd(x, y ) + A,

8x, y 2 X

• (c, A) bilipschitz maps:c 1 d(x, y ) A  d(f (x), f (y ))  cd(x, y )+A,

8x, y 2 X

The large
scalegeometry

Coarse categoryŽiga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions ofunity• Objects: metric spaces;• Morphisms: [equivalence classes of] coarse maps;• Spaces X and Y are coarsely equivalent [X ⇠ls Y ], if

there 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 on
the size of the image:d(f (x), f (y ))  cd(x, y ) + A,

8x, y 2 X

• (c, A) bilipschitz maps:c 1 d(x, y ) A  d(f (x), f (y ))  cd(x, y )+A,

8x, y 2 X

The large
scalegeometry

Coarse categoryŽiga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions ofunity• Objects: metric spaces;• Morphisms: [equivalence classes of] coarse maps;• Spaces X and Y are coarsely equivalent [X ⇠ls Y ], if

there 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 on
the size of the image:d(f (x), f (y ))  cd(x, y ) + A,

8x, y 2 X

• (c, A) bilipschitz maps:c 1 d(x, y ) A  d(f (x), f (y ))  cd(x, y )+A,

8x, y 2 X

The large
scalegeometry

Coarse categoryŽiga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions ofunity• Objects: metric spaces;• Morphisms: [equivalence classes of] coarse maps;• Spaces X and Y are coarsely equivalent [X ⇠ls Y ], if

there 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 on
the size of the image:d(f (x), f (y ))  cd(x, y ) + A,

8x, y 2 X

• (c, A) bilipschitz maps:c 1 d(x, y ) A  d(f (x), f (y ))  cd(x, y )+A,

8x, y 2 X

The large
scalegeometryŽiga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions ofunity• ls-Lipschitz correspond with coarse maps in the usual

setting 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
scalegeometryŽiga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions ofunity• ls-Lipschitz correspond with coarse maps in the usual

setting 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
scalegeometryŽiga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions ofunity• ls-Lipschitz correspond with coarse maps in the usual

setting 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
scalegeometryŽiga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions ofunity• ls-Lipschitz correspond with coarse maps in the usual

setting 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
scalegeometry

The group settingŽiga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions ofunity

G =< g1 , . . . , gn | r1 , . . . > a finitely generated group.
Cayley graph G :• vertices: elements of G ;• edges: [u, v ] , 9i : u = vgi±1 ;• loops in G are ”relations” on the generating set;
• G is geodesic;•

G depends on presentation, its coarse type does not.

The large
scalegeometry

The group settingŽiga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions ofunity

G =< g1 , . . . , gn | r1 , . . . > a finitely generated group.
Cayley graph G :• vertices: elements of G ;• edges: [u, v ] , 9i : u = vgi±1 ;• loops in G are ”relations” on the generating set;
• G is geodesic;•

G depends on presentation, its coarse type does not.

The large
scalegeometry

The group settingŽiga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions ofunity

G =< g1 , . . . , gn | r1 , . . . > a finitely generated group.
Cayley graph G :• vertices: elements of G ;• edges: [u, v ] , 9i : u = vgi±1 ;• loops in G are ”relations” on the generating set;
• G is geodesic;•

G depends on presentation, its coarse type does not.

The large
scalegeometryeŽiga Virk
The idea
The coarse
category
The group
setting

a

a

b

b

Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity

a2

ab

ab

a2

ab

ab

ba

ba

b2

ba

ba

b

2

The large
scalegeometry

Coarse invariants of groupsŽiga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions ofunity

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 has
virtual property
P if there exists a
finite index
subgroup H  Gwith property P.

The large
scalegeometry

Coarse invariants of groupsŽiga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions ofunity

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 has
virtual property
P if there exists a
finite index
subgroup H  Gwith property P.

The large
scalegeometryŽiga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions ofunity

Application to Novikov conjecture by Yu:• Novikov conjecture holds for groups of finite asymptotic

dimension;• Novikov conjecture holds for groups with property A.

Important aspect: coarse embeddings into Hilbert space.

The large
scalegeometryŽiga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions ofunity

Application to Novikov conjecture by Yu:• Novikov conjecture holds for groups of finite asymptotic

dimension;• Novikov conjecture holds for groups with property A.

Important aspect: coarse embeddings into Hilbert space.

The large
scalegeometryŽiga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions ofunity

Application to Novikov conjecture by Yu:• Novikov conjecture holds for groups of finite asymptotic

dimension;• Novikov conjecture holds for groups with property A.

Important aspect: coarse embeddings into Hilbert space.

The large
scalegeometry

Property A can be considered as a variant of the amenability.

Žiga Virk
The idea
The coarse
category
The group
setting

A countable discrete group G is amenable if 8R, " > 0 exists a
finitely supported f 2 `1 (G ) such that:
1
2

Motivation
The
combinatorial
approach
Lipschitz
partitions ofunity||f ||1 = 1;||gf

f ||1 < ", 8g 2 B(e, R).

A countable discrete group G has property A if there exists
1  p < 1 so that 8R, " > 0 exists F : G ! `1 (G ) with thefollowing properties:
1
2
3||Fx ||p = 1, 8x 2 G ;||Fx

Fy ||p < ", 8x, y 2 G , d(x, y ) < R;

9S > 0 : supp Fx ⇢ B(x, S), 8x 2 G .

Free group on two elements has property A but is not
amenable.

The large
scalegeometry

Property A can be considered as a variant of the amenability.

Žiga Virk
The idea
The coarse
category
The group
setting

A countable discrete group G is amenable if 8R, " > 0 exists a
finitely supported f 2 `1 (G ) such that:
1
2

Motivation
The
combinatorial
approach
Lipschitz
partitions ofunity||f ||1 = 1;||gf

f ||1 < ", 8g 2 B(e, R).

A countable discrete group G has property A if there exists
1  p < 1 so that 8R, " > 0 exists F : G ! `1 (G ) with thefollowing properties:
1
2
3||Fx ||p = 1, 8x 2 G ;||Fx

Fy ||p < ", 8x, y 2 G , d(x, y ) < R;

9S > 0 : supp Fx ⇢ B(x, S), 8x 2 G .

Free group on two elements has property A but is not
amenable.

The large
scalegeometry

Property A can be considered as a variant of the amenability.

Žiga Virk
The idea
The coarse
category
The group
setting

A countable discrete group G is amenable if 8R, " > 0 exists a
finitely supported f 2 `1 (G ) such that:
1
2

Motivation
The
combinatorial
approach
Lipschitz
partitions ofunity||f ||1 = 1;||gf

f ||1 < ", 8g 2 B(e, R).

A countable discrete group G has property A if there exists
1  p < 1 so that 8R, " > 0 exists F : G ! `1 (G ) with thefollowing properties:
1
2
3||Fx ||p = 1, 8x 2 G ;||Fx

Fy ||p < ", 8x, y 2 G , d(x, y ) < R;

9S > 0 : supp Fx ⇢ B(x, S), 8x 2 G .

Free group on two elements has property A but is not
amenable.

The large
scalegeometry

Property A can be considered as a variant of the amenability.

Žiga Virk
The idea
The coarse
category
The group
setting

A countable discrete group G is amenable if 8R, " > 0 exists a
finitely supported f 2 `1 (G ) such that:
1
2

Motivation
The
combinatorial
approach
Lipschitz
partitions ofunity||f ||1 = 1;||gf

f ||1 < ", 8g 2 B(e, R).

A countable discrete group G has property A if there exists
1  p < 1 so that 8R, " > 0 exists F : G ! `1 (G ) with thefollowing properties:
1
2
3||Fx ||p = 1, 8x 2 G ;||Fx

Fy ||p < ", 8x, y 2 G , d(x, y ) < R;

9S > 0 : supp Fx ⇢ B(x, S), 8x 2 G .

Free group on two elements has property A but is not
amenable.

The large
scalegeometryŽiga Virk
The idea

Property A can be considered as a weakening of the finite
asymptotic dimension.

The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions ofunity

A discrete metric space of bounded geometry has property A,
if 8R, " > 0, 9S > 0 and a family of finite nonempty subsets{Ax }x2X of X ⇥ N so that:
1
2

|A

A |

if x, y 2 X , d(x, y )  R then |Axx \Ayy | < ";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 onto
X has at most n + 1 elements, 8x 2 X , 8R, " > 0.

The large
scalegeometryŽiga Virk
The idea

Property A can be considered as a weakening of the finite
asymptotic dimension.

The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions ofunity

A discrete metric space of bounded geometry has property A,
if 8R, " > 0, 9S > 0 and a family of finite nonempty subsets{Ax }x2X of X ⇥ N so that:
1
2

|A

A |

if x, y 2 X , d(x, y )  R then |Axx \Ayy | < ";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 onto
X has at most n + 1 elements, 8x 2 X , 8R, " > 0.

The large
scalegeometry

The combinatorial approachŽiga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions ofunity

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 : graphwith added edges. For
vertices u, v , d(u, v ) = 2 the edge [u, v ] is added;• A( ) of a graph : A( ) is a complex ofso thatis a
simplex of A( ) whenever [v , w ] belongs to for allv, w 2 ;• motivation: Rips graphs of a space.

The large
scalegeometry

The combinatorial approachŽiga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions ofunity

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 : graphwith added edges. For
vertices u, v , d(u, v ) = 2 the edge [u, v ] is added;• A( ) of a graph : A( ) is a complex ofso thatis a
simplex of A( ) whenever [v , w ] belongs to for allv, w 2 ;• motivation: Rips graphs of a space.

The large
scalegeometry

Approximation by the system ofgraphsŽiga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions ofunity

Objects: Coarse graphs.
A coarse graph is a direct sequence {V1 ! V2 ! . . .} ofgraphs Vn and short maps in,m : Vn ! Vm for all n  m suchthat
1

in,n = id for all n

2

in,k = im,k3

for every n
is short.

1,

in,m for all n  m  k,

1 there is m > n so that in,m : A(Vn ) ! Vm

The large
scalegeometry

Approximation by the system ofgraphsŽiga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions ofunity

Objects: Coarse graphs.
A coarse graph is a direct sequence {V1 ! V2 ! . . .} ofgraphs Vn and short maps in,m : Vn ! Vm for all n  m suchthat
1

in,n = id for all n

2

in,k = im,k3

for every n
is short.

1,

in,m for all n  m  k,

1 there is m > n so that in,m : A(Vn ) ! Vm

The large
scalegeometryŽiga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions ofunity

Morphisms: equivalence classes of pre-morphisms.
i1,2

i2,3

j1,2

j2,3

Suppose V = {V1 ! V2 ! . . .} and W = {W1 ! W2 ! . . .}are two coarse graphs. A pre-morphism F : V ! W: afunction nF : N ! N, and short maps fk : Vk ! WnF (k) so thatfor every k 1 there is m nF (k + 1) resulting in
jnF (k),m fk ⇠ls jnF (k+1),m fk+1 ik,k+1 .
W1

j1,2

W2

j2,3

···

j

WnF (1)
f2

f1
V1

i1,2

V2

i2,3

V3

i3,4

···

j

···

j
f1

WnF (2)j···j

WnF (3) ·The large
scalegeometryŽiga Virk

W1

j1,2

W2

The idea

Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity

···

j

WnF (1)
f2

f1

The coarse
category
The group
setting

j2,3

V1

i1,2

V2

i2,3

V3

i3,4

j

···

j

WnF (2)j···j

WnF (3) ·f1···

Two pre-morphisms F , G : V ! W are considered to beequivalent if for every k there is m max{nF (k), nG (k)} sothat jnF (k),m fk ⇠ls jnG (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 exist
pre-morphisms F : V ! W and G : W ! V such that G F isequivalent to idV and F G is equivalent to idW .

The large
scalegeometryŽiga Virk

W1

j1,2

W2

The idea

Motivation
The
combinatorial
approach
Lipschitz
partitions of
unity

···

j

WnF (1)
f2

f1

The coarse
category
The group
setting

j2,3

V1

i1,2

V2

i2,3

V3

i3,4

j

···

j

WnF (2)j···j

WnF (3) ·f1···

Two pre-morphisms F , G : V ! W are considered to beequivalent if for every k there is m max{nF (k), nG (k)} sothat jnF (k),m fk ⇠ls jnG (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 exist
pre-morphisms F : V ! W and G : W ! V such that G F isequivalent to idV and F G is equivalent to idW .

The large
scalegeometryŽiga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions ofunity

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 RipsGr i↵ d(u, v )  r .
• Rips graph of an increasing sequence of uniformly bounded

covers Un for which Lebesgue number ! 1 :1
2

maps i are identity;[u, v ] is an edge in RipsGUn i↵ u and v lie in the sameelement of Un .

If V = {V1 ! V2 ! . . .} is a coarse graph of (X , dX ) and
W = {W1 ! W2 ! . . .} is a coarse graph of (Y , dY ), thenthere is a natural bijection between bornologous maps from X
to Y and morphisms from V to W.
The description is sufficient for the formulation of the coarse
type but not for some other invariants.

The large
scalegeometryŽiga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions ofunity

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 RipsGr i↵ d(u, v )  r .
• Rips graph of an increasing sequence of uniformly bounded

covers Un for which Lebesgue number ! 1 :1
2

maps i are identity;[u, v ] is an edge in RipsGUn i↵ u and v lie in the sameelement of Un .

If V = {V1 ! V2 ! . . .} is a coarse graph of (X , dX ) and
W = {W1 ! W2 ! . . .} is a coarse graph of (Y , dY ), thenthere is a natural bijection between bornologous maps from X
to Y and morphisms from V to W.
The description is sufficient for the formulation of the coarse
type but not for some other invariants.

The large
scalegeometryŽiga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions ofunity

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 RipsGr i↵ d(u, v )  r .
• Rips graph of an increasing sequence of uniformly bounded

covers Un for which Lebesgue number ! 1 :1
2

maps i are identity;[u, v ] is an edge in RipsGUn i↵ u and v lie in the sameelement of Un .

If V = {V1 ! V2 ! . . .} is a coarse graph of (X , dX ) and
W = {W1 ! W2 ! . . .} is a coarse graph of (Y , dY ), thenthere is a natural bijection between bornologous maps from X
to Y and morphisms from V to W.
The description is sufficient for the formulation of the coarse
type but not for some other invariants.

The large
scalegeometryŽiga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions ofunity

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 RipsGr i↵ d(u, v )  r .
• Rips graph of an increasing sequence of uniformly bounded

covers Un for which Lebesgue number ! 1 :1
2

maps i are identity;[u, v ] is an edge in RipsGUn i↵ u and v lie in the sameelement of Un .

If V = {V1 ! V2 ! . . .} is a coarse graph of (X , dX ) and
W = {W1 ! W2 ! . . .} is a coarse graph of (Y , dY ), thenthere is a natural bijection between bornologous maps from X
to Y and morphisms from V to W.
The description is sufficient for the formulation of the coarse
type but not for some other invariants.

The large
scalegeometry

Approximation by the system ofcomplexesŽiga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions ofunity

Objects: Coarse complexes.
A coarse simplicial complex is a direct sequence{V1 ! V2 ! . . .} of simplicial complexes Vn and simplicial
maps in,m : Vn ! Vm for all n  m such that1

in,n = id for all n

2

in,k = im,k3

for every n
is short.

1,

in,m for all n  m  k,

1 there is m > n so that in,m : A(Vn ) ! Vm

The large
scalegeometry

Approximation by the system ofcomplexesŽiga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions ofunity

Objects: Coarse complexes.
A coarse simplicial complex is a direct sequence{V1 ! V2 ! . . .} of simplicial complexes Vn and simplicial
maps in,m : Vn ! Vm for all n  m such that1

in,n = id for all n

2

in,k = im,k3

for every n
is short.

1,

in,m for all n  m  k,

1 there is m > n so that in,m : A(Vn ) ! Vm

The large
scalegeometryŽiga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions ofunity

Morphisms: equivalence classes of pre-morphisms.
i1,2

i2,3

j1,2

j2,3

Suppose V = {V1 ! V2 ! . . .} and W = {W1 ! W2 ! . . .}are two coarse coarse simplicial complexes. A pre-morphism
F : V ! W: a function nF : N ! N, and simplicial maps
fk : Vk ! WnF (k) so that for every k 1 there ism nF (k + 1) resulting in
jnF (k),m fk ⇠ls jnF (k+1),m fk+1 ik,k+1 .
j1,2
j2,3
j
j
j
j
j
W1
W2
···
WnF (1)···WnF (2)···WnF (3) ·
f2

f1
V1

i1,2

V2

i2,3

V3

i3,4

···

f1

The large
scalegeometryŽiga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions ofunity

Associating a coarse simplicial complex to the metric space.
• Rips complexes Ripsr for r ! 1:
1 maps i are identity;
2 [u0 , u1 , u2 , . . . , uk ] is a simplex in Ripsr i↵
Diam{u0 , u1 , u2 , . . . , uk }  r .
• Rips complex of an increasing sequence of uniformly

bounded covers Un for which Lebesgue number ! 1 :1
2

maps i are identity;[u0 , u1 , u2 , . . . , uk ] is a simplex in RipsUn i↵[u0 , u1 , u2 , . . . , uk ] is a subset of some element of Un .

The large
scalegeometryŽiga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions ofunity

Associating a coarse simplicial complex to the metric space.
• Rips complexes Ripsr for r ! 1:
1 maps i are identity;
2 [u0 , u1 , u2 , . . . , uk ] is a simplex in Ripsr i↵
Diam{u0 , u1 , u2 , . . . , uk }  r .
• Rips complex of an increasing sequence of uniformly

bounded covers Un for which Lebesgue number ! 1 :1
2

maps i are identity;[u0 , u1 , u2 , . . . , uk ] is a simplex in RipsUn i↵[u0 , u1 , u2 , . . . , uk ] is a subset of some element of Un .

The large
scalegeometryŽiga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions ofunity

Suppose Un is a sequence of uniformly bounded covers of X
such that Un 1 is a star refinement of Un for each n 1. Thenthe sequence N (U1 ) ! N (U2 ) ! . . . of nerves of covers Un
forms a coarse simplicial complex if in,n+1 (U) contains the starst(U, Un ) for each U 2 Un . Any such coarse complex will bedenoted by Čech⇤ (X ) and called a coarse Čech complex of X .
If V = {V1 ! V2 ! . . .} is a coarse simplicial complex of(X , dX ) and W = {W1 ! W2 ! . . .} is a coarse simplicial
complex of (Y , dY ), then there is a natural bijection between
bornologous maps from X to Y and morphisms from V to W.

The large
scalegeometryŽiga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions ofunity

Suppose Un is a sequence of uniformly bounded covers of X
such that Un 1 is a star refinement of Un for each n 1. Thenthe sequence N (U1 ) ! N (U2 ) ! . . . of nerves of covers Un
forms a coarse simplicial complex if in,n+1 (U) contains the starst(U, Un ) for each U 2 Un . Any such coarse complex will bedenoted by Čech⇤ (X ) and called a coarse Čech complex of X .
If V = {V1 ! V2 ! . . .} is a coarse simplicial complex of(X , dX ) and W = {W1 ! W2 ! . . .} is a coarse simplicial
complex of (Y , dY ), then there is a natural bijection between
bornologous maps from X to Y and morphisms from V to W.

The large
scalegeometry

Asymptotic dimension

Žiga Virk
The idea
The coarse
category

Simplicial maps f , g : K ! L between simplicial complexes arecontiguous if for every simplexof K , f ( ) [ g ( ) is
contained in some simplex of L.

The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions ofunity

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 im,k factors contiguously throughan n-dimensional simplicial complex.
im,k
···

Km

···

L(n)

Kk

···

The large
scalegeometry

Asymptotic dimension

Žiga Virk
The idea
The coarse
category

Simplicial maps f , g : K ! L between simplicial complexes arecontiguous if for every simplexof K , f ( ) [ g ( ) is
contained in some simplex of L.

The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions ofunity

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 im,k factors contiguously throughan n-dimensional simplicial complex.
im,k
···

Km

···

L(n)

Kk

···

The large
scalegeometry

Property A

Žiga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions ofunity

A coarse simplicial complex K = {K1 ! K2 ! . . .} has
Property A if for each k 1 and each ✏ > 0 there is n > k
and a function f : |Kk |m ! |Kn |m such that f is contiguous toik,n : |Kk | ! |Kn | and the diameter of f (| |) is at most ✏ foreach simplexof Kk .
im,k
···

Kk

···

Kn

|Kk |mf|Kn |m···The large
scalegeometry

Coarse simple connectivity

Žiga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach

A metric space X is coarsely k-connected if for each r there
exists R r so that the mapping |Ripsr (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 H1 (X ) is
uniformly generated and X is of asymptotic dimension 1;

3

An application of the previous item: finitely presented
groups of asymptotic dimension 1 are virtually free;4

If the fundamental group of a compact, connected, locallyconnected metric space is countable, then it is finitely
presented.

Lipschitz
partitions ofunityThe large
scalegeometry

Coarse simple connectivity

Žiga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach

A metric space X is coarsely k-connected if for each r there
exists R r so that the mapping |Ripsr (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 H1 (X ) is
uniformly generated and X is of asymptotic dimension 1;

3

An application of the previous item: finitely presented
groups of asymptotic dimension 1 are virtually free;4

If the fundamental group of a compact, connected, locallyconnected metric space is countable, then it is finitely
presented.

Lipschitz
partitions ofunityThe large
scalegeometry

Coarse simple connectivity

Žiga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach

A metric space X is coarsely k-connected if for each r there
exists R r so that the mapping |Ripsr (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 H1 (X ) is
uniformly generated and X is of asymptotic dimension 1;

3

An application of the previous item: finitely presented
groups of asymptotic dimension 1 are virtually free;4

If the fundamental group of a compact, connected, locallyconnected metric space is countable, then it is finitely
presented.

Lipschitz
partitions ofunityThe large
scalegeometry

Coarse simple connectivity

Žiga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach

A metric space X is coarsely k-connected if for each r there
exists R r so that the mapping |Ripsr (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 H1 (X ) is
uniformly generated and X is of asymptotic dimension 1;

3

An application of the previous item: finitely presented
groups of asymptotic dimension 1 are virtually free;4

If the fundamental group of a compact, connected, locallyconnected metric space is countable, then it is finitely
presented.

Lipschitz
partitions ofunityThe large
scalegeometry

Coarse simple connectivity

Žiga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach

A metric space X is coarsely k-connected if for each r there
exists R r so that the mapping |Ripsr (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 H1 (X ) is
uniformly generated and X is of asymptotic dimension 1;

3

An application of the previous item: finitely presented
groups of asymptotic dimension 1 are virtually free;4

If the fundamental group of a compact, connected, locallyconnected metric space is countable, then it is finitely
presented.

Lipschitz
partitions ofunityThe large
scalegeometryŽiga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions ofunity

If the fundamental group of a compact, connected, locallyconnected metric space X is countable, then it is finitely
presented:e;
1 ⇡1 (X ) countable ) 9X

e geodesic;
X geodesic ) Xe ) [Švarc-Milnor Lemma] ⇡1 (X ) finitely
3 ⇡1 (X ) acts on Xe;generated and ⇡1 (X ) 'ls X
245e coarsely simply connected ) ⇡1 (X ) coarsely simplyX
connected;⇡1 (X ) finitely presented.

The large
scalegeometryŽiga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions ofunity

If the fundamental group of a compact, connected, locallyconnected metric space X is countable, then it is finitely
presented:e;
1 ⇡1 (X ) countable ) 9X

e geodesic;
X geodesic ) Xe ) [Švarc-Milnor Lemma] ⇡1 (X ) finitely
3 ⇡1 (X ) acts on Xe;generated and ⇡1 (X ) 'ls X
245e coarsely simply connected ) ⇡1 (X ) coarsely simplyX
connected;⇡1 (X ) finitely presented.

The large
scalegeometryŽiga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions ofunity

If the fundamental group of a compact, connected, locallyconnected metric space X is countable, then it is finitely
presented:e;
1 ⇡1 (X ) countable ) 9X

e geodesic;
X geodesic ) Xe ) [Švarc-Milnor Lemma] ⇡1 (X ) finitely
3 ⇡1 (X ) acts on Xe;generated and ⇡1 (X ) 'ls X
245e coarsely simply connected ) ⇡1 (X ) coarsely simplyX
connected;⇡1 (X ) finitely presented.

The large
scalegeometryŽiga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions ofunity

If the fundamental group of a compact, connected, locallyconnected metric space X is countable, then it is finitely
presented:e;
1 ⇡1 (X ) countable ) 9X

e geodesic;
X geodesic ) Xe ) [Švarc-Milnor Lemma] ⇡1 (X ) finitely
3 ⇡1 (X ) acts on Xe;generated and ⇡1 (X ) 'ls X
245e coarsely simply connected ) ⇡1 (X ) coarsely simplyX
connected;⇡1 (X ) finitely presented.

The large
scalegeometryŽiga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions ofunity

If the fundamental group of a compact, connected, locallyconnected metric space X is countable, then it is finitely
presented:e;
1 ⇡1 (X ) countable ) 9X

e geodesic;
X geodesic ) Xe ) [Švarc-Milnor Lemma] ⇡1 (X ) finitely
3 ⇡1 (X ) acts on Xe;generated and ⇡1 (X ) 'ls X
245e coarsely simply connected ) ⇡1 (X ) coarsely simplyX
connected;⇡1 (X ) finitely presented.

The large
scalegeometryŽiga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions ofunity

If the fundamental group of a compact, connected, locallyconnected metric space X is countable, then it is finitely
presented:e;
1 ⇡1 (X ) countable ) 9X

e geodesic;
X geodesic ) Xe ) [Švarc-Milnor Lemma] ⇡1 (X ) finitely
3 ⇡1 (X ) acts on Xe;generated and ⇡1 (X ) 'ls X
245e coarsely simply connected ) ⇡1 (X ) coarsely simplyX
connected;⇡1 (X ) finitely presented.

The large
scalegeometry

Partitions of unityŽiga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions ofunity

Given > 0 and a simplicial complex K , f : X ! K is apartition of unity if it is ( , ) Lipschitz and{f 1 (st(v ))}v 2K (0) is a uniformly bounded cover with Lebesgue
number 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
scalegeometry

Partitions of unityŽiga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions ofunity

Given > 0 and a simplicial complex K , f : X ! K is apartition of unity if it is ( , ) Lipschitz and{f 1 (st(v ))}v 2K (0) is a uniformly bounded cover with Lebesgue
number 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
scalegeometry

Partitions of unityŽiga Virk
The idea
The coarse
category
The group
setting
Motivation
The
combinatorial
approach
Lipschitz
partitions ofunity

Given > 0 and a simplicial complex K , f : X ! K is apartition of unity if it is ( , ) Lipschitz and{f 1 (st(v ))}v 2K (0) is a uniformly bounded cover with Lebesgue
number 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.