Post on 10-Feb-2017
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.