# What is a Dehn function? “What is...?” Seminar

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What is a Dehn function?

Cornell, September 9, 2009

What is...? Seminar

Tim Riley

Euclidean space enjoys a quadratic isoperimetric function.

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

Xa loop in a simply connected space

is defined by

AreaX(l) = sup{Area() | () l}.

AreaX : [0,) [0,]

Z3R3

2-cellsArea() = #

1(K) =

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

The presentation 2-complex of :P

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

K =

r1 r2 r3

r4 rna1

a2 a4

am

a3

a b c

d

a

a

b

bc

c

d

d

Z2a, b | a1b1ab a, b, c, d | a1b1ab c1d1cd

b

a

a b

b

a

a, b | b1aba2

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 . ba1ca1bcb1ca1b1c1bc1b2acac1a1c1aba

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 NK

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

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

PArea()

f g when such that .C > 0 n > 0, f(n) Cg(Cn + C) + Cn + Cf g when and .f g g f

a, b | a1b1ab a, b, c, d | a1b1ab c1d1cd

a, b | b1aba2

Area(n) n Area(n) n2

Area(n) 2n

van Kampens 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

w = 1

F (A)w =N

i=1

ui1ri

iui i = 1 uiri R

w

r11

r22r3

3

r44

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 = { > 0 | n n }

hyperbolicgroups

10 2 43 5 6

The closure of isIP

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

Theorem. (SapirBirgetRips.) 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 .

> 4 c > 0

c22cm

IP IP

c222cm c > 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 | b1aba2

via and

GL2(Q)

a

1 10 1

b

1/2 00 1

Example:

Dehn function is a geometric timecomplexity measure

Big Dehn functions

exp(n)a, b | b1ab = a2

a, b, c | b1ab = a2, c1bc = b2

a, b, c, d | b1ab = a2, c1bc = b2, d1cd = c2

exp(exp(n))

...

a, b | (b1ab)1a(b1ab) = a2

...

exp(. . . (exp (n)) . . .)log n

exp(exp(exp(n)))

The group generated by

a1, . . . , ak, p, t

subject to

t1ait = aiai1t1a1t = a1[p, ait] = 1 i

i > 1for all

for all

has Dehn function like the -th of Ackermanns functions

n Ak(n) .

k

Hydra Groups

There are finitely presented groups with unsolvable Word Problem (NovikovBoone).

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