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

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Transcript of What is a Dehn function? “What is...?” Seminar

  • 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

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    b

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

    b

    a

    a b

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    a, b | b1aba2

    ba

    b

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    a a

    b( )

  • Z3

    a b

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    a ac

    c

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    cc

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    a ab

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    aa a a

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    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.