What is a Dehn function?

Cornell, September 9, 2009

“What is...?” Seminar

Tim Riley

Euclidean space “enjoys a quadratic isoperimetric function.”

ρ

Area(ρ) is the infimum of the areas of discs spanning . ρ

Xa loop in a simply connected space

is defined by

AreaX(l) = sup{Area(ρ) | �(ρ) ≤ l}.

AreaX : [0,∞)→ [0,∞]

Z3R3

2-cellsArea(∆) = #

∆

π1(K) = Γ

a finite presentation of a group ΓP = �a1, . . . , am | r1, . . . , rn�

The presentation 2-complex of :P

�KThe universal cover is the Cayley 2-complex of .P�K(1)Its 1-skeleton is the Cayley graph of .P

K =

r1 r2 r3

r4 rn

a1

a2 a4

am

a3

a bc

d

a

a

b

bc

c

d

d

Z2

�a, b | a−1b−1ab�� a, b, c, d | a−1b−1ab c−1d−1cd �

b

a

a b

b

a

�a, b | b−1aba−2�

ba

b

a

a a

b( )

Z3

∗∗

a b

c

a

a ac

c

cb

bb

c

a

b

c

a

b

a

cc

c c c

ca a a

a ab

b

b

bb

b b

b

b

aa a a

aa

a

c

c b

b

b

b

c c cb

b

a

A van Kampen diagram for . ba−1ca−1bcb−1ca−1b−1c−1bc−1b−2acac−1a−1c−1aba

� a, b, c | �=

A van Kampen diagram for a word is a finite planar contractible 2-complex with edges directed and labelled so that around each 2-cell one reads a defining relation and around one reads .

w∆

w∂∆

The Filling Theorem. If is a finite presentation of the fundamental group of a closed Riemannian manifold then

PM

.AreaP � AreafM

The Dehn function of a finite presentation with Cayley 2-complex is P

AreaP : N→ N�K

edge-loops in with .AreaP(n) = max{Area(ρ) | ρ �K �(ρ) ≤ n}

Area(ρ)For an edge-loop in the Cayley 2-complex of a finite presentation , is the minimum of over all van Kampen diagrams spanning .

PArea(∆)

ρ

ρ

f � g when such that .∃C > 0 ∀n > 0, f(n) ≤ Cg(Cn + C) + Cn + C

f � g when and .f � g g � f

�a, b | a−1b−1ab�� a, b, c, d | a−1b−1ab c−1d−1cd �

�a, b | b−1aba−2�

Area(n) � n Area(n) � n2

Area(n) � 2n

van Kampen’s Lemma. For a group with finite presentation , and a word , the following are equivalent. w

(i) in ,

(ii) in for some , , words ,

(iii) admits a van Kampen diagram.

�A | R�Γ

w = 1 Γ

F (A)w =N�

i=1

ui−1ri

εiui εi = ±1 uiri ∈ R

w

r1ε1

r2ε2r3

ε3

r4ε4

u1

u2u3

u4

w

Technicalities hidden!

Moreover, is the minimal as occurring in (ii).Area(w) N

Examples of Dehn functions

• finite groups: ≤ Cn

• free groups: ≤ Cn

• hyperbolic groups groups with Dehn function ≤ Cn:=

• 3-dimensional integral Heisenberg group: � n3

• class free nilpotent group on letters:c 2 � nc+1

• finitely generated abelian groups: ≤ Cn2

is a Dehn functionIP = {α > 0 | n �→ nα � }

hyperbolicgroups

10 2 43 5 6

The closure of isIP

Gromov, Bowditch, N. Brady, Bridson, Olshanskii, Sapir...

Theorem. (Sapir–Birget–Rips.) If is such that and a Turing Machine calculating the first digits of the decimal expansion of in time then . Conversely, ifthen there exists a Turing Machine that calculates the first digits in time for some constant .

α > 4 ∃c > 0

α ≤ c22cm

α ∈ IP α ∈ IP

≤ c222cmc > 0

m

m

Dehn functions and the Word Problem

For a finitely presented group, t.f.a.e.:

• the word problem is solvable,

• the Dehn function is recursive,

• the Dehn function is bounded from above by a recursive function.

But groups with large Dehn function may have efficient solutions to their word problem.

�a, b | b−1aba−2�

via and

≤ GL2(Q)

a �→�

1 10 1

�b �→

�1/2 00 1

�

Example:

Dehn function “is” a geometric time–complexity measure

Big Dehn functions

exp(n)�a, b | b−1ab = a2�

�a, b, c | b−1ab = a2, c−1bc = b2�

�a, b, c, d | b−1ab = a2, c−1bc = b2, d−1cd = c2�

exp(exp(n))

...

�a, b | (b−1ab)−1a(b−1ab) = a2�

...

exp(. . . (exp� �� �(n)) . . .)

log n

exp(exp(exp(n)))

The group generated by

a1, . . . , ak, p, t

subject to

t−1ait = aiai−1

t−1a1t = a1

[p, ait] = 1 i

i > 1for all

for all

has Dehn function like the -th of Ackermann’s functions

n �→ Ak(n) .

k

Hydra Groups

There are finitely presented groups with unsolvable Word Problem (Novikov–Boone).

These have Dehn function which cannot be bounded from above by a recursive function.