What Is a Dehn Function? “What Is...?” Seminar

Tim Riley

“What is...?” Seminar

Cornell, September 9, 2009 Euclidean space “enjoys a quadratic isoperimetric function.” ρ a loop in a simply connected space X

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

Area : [0, ) [0, ] is deﬁned by X ∞ → ∞ AreaX (l) = sup Area(ρ) (ρ) l . { | ≤ } 33 ZR

∆

Area(∆) = # 2-cells = a1, . . . , am r1, . . . , rn a ﬁnite presentation of a group Γ P | The presentation 2-complex of : P

r1 r2 r3 a3 a2 a4 K = a1 r4 rn am

π1(K) = Γ

The universal cover K is the Cayley 2-complex of . P Its 1-skeleton K (1 ) is the Cayley graph of . P 1 1 1 1 1 1 a, b, c, d a− b− ab c− d− cd a, b a− b− ab | | 2 Z

a a a d b b b a c b b c a d b a d c

1 2 a, b b− aba− |

a a a b ( b b ) a a b c 3 = a, b, c b b c c a a Z | a b c c a b c b c c a a a b c b b b c c c ca b a b a b a a a c c b c b b b a b a c a b b a a ∗ a ∗ b 1 1 1 1 1 1 1 2 1 1 1 A van Kampen diagram for ba − ca − bcb − ca − b − c − bc − b − acac − a − c − aba .

A van Kampen diagram for a word w 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 . For an edge-loop ρ in the Cayley 2-complex of a finite presentation , Area( ρ ) is the minimum of Area( ∆ ) over all van Kampen diagramsP spanning ρ .

The Dehn function Area : N N of a ﬁnite presentation P → with Cayley 2-complex K is P Area (n) = max Area( ρ) edge-loops ρ in K with ( ρ ) n . P { | ≤ } The Filling Theorem. If is a ﬁnite presentation of the fundamental group of a closedP Riemannian manifold M then Area Area . P M f

f g when C> 0 such that n> 0 ,f ( n ) Cg ( Cn + C )+ Cn + C . ∃ ∀ ≤ f g when f g and g f . 1 1 1 1 1 1 a, b, c, d a− b− ab c− d− cd a, b a− b− ab | |

Area(n) n Area(n) n2

1 2 a, b b− aba− |

Area(n) 2n van Kampen’s Lemma. For a group Γ with ﬁnite presentation , and a word w , the following are equivalent. A|R (i) w =1 in Γ , N 1 εi (ii) w = u i − r i u i in F ( ) for some r i , ε i = 1 , words u , A ∈ R ± i i=1 (iii) w admits a van Kampen diagram.

ε3 ε2 r3 r2 w

ε4 r4 u3 u2 ε1 r1 u4 u1

Technicalities hidden!

Moreover, Area( w ) is the minimal N as occurring in (ii). Examples of Dehn functions • ﬁnite groups: Cn ≤ • free groups: Cn ≤ • hyperbolic groups := groups with Dehn function Cn ≤ • ﬁnitely generated abelian groups: Cn2 ≤ • 3-dimensional integral Heisenberg group: n3 class c free nilpotent group on 2 letters: nc+1 • IP = α > 0 n nα is a Dehn function { | → }

The closure of IP is

0 1 2 3 4 5 6

hyperbolic groups

Gromov, Bowditch, N. Brady, Bridson, Olshanskii, Sapir... Theorem. (Sapir–Birget–Rips.) If α > 4 is such that c> 0 and a Turing Machine calculating the first m digits of the decimal∃ cm expansion of α in time c 2 2 then α IP . Conversely, if α IP then there exists a Turing≤ Machine that calculates∈ the first m digits∈ in 2cm time c 2 2 for some constant c> 0 . ≤ Dehn functions and the Word Problem For a finitely presented group, t.f.a.e.: • the word problem is solvable, Dehn function “is” a geometric the Dehn function is recursive, • time–complexity • the Dehn function is bounded from measure above by a recursive function.

But groups with large Dehn function may have efﬁcient solutions to their word problem. Example: 1 2 a, b b− aba− GL2(Q) | ≤ 11 1/20 via a and b → 01 → 01 Big Dehn functions

1 2 a, b b− ab = a exp(n) | 1 2 1 2 a, b, c b− ab = a ,c− bc = b exp(exp(n)) |

1 2 1 2 1 2 a, b, c, d b− ab = a ,c− bc = b ,d− cd = c exp(exp(exp(n))) | . . . .

1 1 1 2 a, b (b− ab)− a(b− ab)=a exp(...(exp(n)) ...) | log n Hydra Groups

The group generated by

a1,...,ak, p, t subject to

1 t− ait = aiai 1 for all i>1 − 1 t− a1t = a1

[p, ait]=1 for all i has Dehn function like the k -th of Ackermann’s functions n A (n) . → k 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.