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What is a group?
Scott Harper
Postgraduate Seminar
2nd November 2018
What is a group?
It depends who you ask.
Groups are viewed from many different perspectives by a diversity ofpeople with a wide range of differing interests. Sometimes two seeminglyunrelated properties of a group, coming from two radically distinctviewpoints, miraculously happen to be equivalent. I’ll tell you about someof the different perspectives on group theory and then focus on one casewhere a surprising bridge was built, the Muller–Schupp theorem.
I expect the audience to arrive knowing what a group is and leave doubtingthat they ever did.
What is a group?
It depends who you ask.
Groups are viewed from many different perspectives by a diversity ofpeople with a wide range of differing interests. Sometimes two seeminglyunrelated properties of a group, coming from two radically distinctviewpoints, miraculously happen to be equivalent. I’ll tell you about someof the different perspectives on group theory and then focus on one casewhere a surprising bridge was built, the Muller–Schupp theorem.
I expect the audience to arrive knowing what a group is and leave doubtingthat they ever did.
Permutation group
〈(1 2 3 4), (1 4)(2 3)〉
Representation theory⟨(0 −11 0
),(−1 00 1
)⟩
Combinatorialgroup theory
〈a, b | a4 = b2 = 1, ba = a−1b〉
D8
symmetries of a square
ab
Abstract group× 1 a a2 a3 b ab a2b a3b1 1 a a2 a3 b ab a2b a3ba a a2 a3 1 ab a2b a3b ba2 a2 a3 1 a a2b a3b b a2ba3 a3 1 a a2 a3b b ab a2bb b ab3 a2b ab 1 a3 a2 aab ab b a3b a2b a 1 a3 a2
a2b a2b ab b a3b a2 a 1 a3
a3b a3b a2b ab b a3 a2 a 1
Geometricgroup theory
1
aa2
a3
ba3b
a2b ab
Permutation group
〈(1 2 3 4), (1 4)(2 3)〉
Representation theory⟨(0 −11 0
),(−1 00 1
)⟩
Combinatorialgroup theory
〈a, b | a4 = b2 = 1, ba = a−1b〉
D8
symmetries of a square
ab
Abstract group× 1 a a2 a3 b ab a2b a3b1 1 a a2 a3 b ab a2b a3ba a a2 a3 1 ab a2b a3b ba2 a2 a3 1 a a2b a3b b a2ba3 a3 1 a a2 a3b b ab a2bb b ab3 a2b ab 1 a3 a2 aab ab b a3b a2b a 1 a3 a2
a2b a2b ab b a3b a2 a 1 a3
a3b a3b a2b ab b a3 a2 a 1
Geometricgroup theory
1
aa2
a3
ba3b
a2b ab
Permutation group
〈(1 2 3 4), (1 4)(2 3)〉
Representation theory⟨(0 −11 0
),(−1 00 1
)⟩
Combinatorialgroup theory
〈a, b | a4 = b2 = 1, ba = a−1b〉
D8
symmetries of a square
ab
Abstract group× 1 a a2 a3 b ab a2b a3b1 1 a a2 a3 b ab a2b a3ba a a2 a3 1 ab a2b a3b ba2 a2 a3 1 a a2b a3b b a2ba3 a3 1 a a2 a3b b ab a2bb b ab3 a2b ab 1 a3 a2 aab ab b a3b a2b a 1 a3 a2
a2b a2b ab b a3b a2 a 1 a3
a3b a3b a2b ab b a3 a2 a 1
Geometricgroup theory
1
aa2
a3
ba3b
a2b ab
Permutation group
〈(1 2 3 4), (1 4)(2 3)〉
Representation theory⟨(0 −11 0
),(−1 00 1
)⟩
Combinatorialgroup theory
〈a, b | a4 = b2 = 1, ba = a−1b〉
D8
symmetries of a square
ab
Abstract group× 1 a a2 a3 b ab a2b a3b1 1 a a2 a3 b ab a2b a3ba a a2 a3 1 ab a2b a3b ba2 a2 a3 1 a a2b a3b b a2ba3 a3 1 a a2 a3b b ab a2bb b ab3 a2b ab 1 a3 a2 aab ab b a3b a2b a 1 a3 a2
a2b a2b ab b a3b a2 a 1 a3
a3b a3b a2b ab b a3 a2 a 1
Geometricgroup theory
1
aa2
a3
ba3b
a2b ab
Permutation group
〈(1 2 3 4), (1 4)(2 3)〉
Representation theory⟨(0 −11 0
),(−1 00 1
)⟩
Combinatorialgroup theory
〈a, b | a4 = b2 = 1, ba = a−1b〉
D8
symmetries of a square
a
b
Abstract group× 1 a a2 a3 b ab a2b a3b1 1 a a2 a3 b ab a2b a3ba a a2 a3 1 ab a2b a3b ba2 a2 a3 1 a a2b a3b b a2ba3 a3 1 a a2 a3b b ab a2bb b ab3 a2b ab 1 a3 a2 aab ab b a3b a2b a 1 a3 a2
a2b a2b ab b a3b a2 a 1 a3
a3b a3b a2b ab b a3 a2 a 1
Geometricgroup theory
1
aa2
a3
ba3b
a2b ab
Permutation group
〈(1 2 3 4), (1 4)(2 3)〉
Representation theory⟨(0 −11 0
),(−1 00 1
)⟩
Combinatorialgroup theory
〈a, b | a4 = b2 = 1, ba = a−1b〉
D8
symmetries of a square
a
b
Abstract group× 1 a a2 a3 b ab a2b a3b1 1 a a2 a3 b ab a2b a3ba a a2 a3 1 ab a2b a3b ba2 a2 a3 1 a a2b a3b b a2ba3 a3 1 a a2 a3b b ab a2bb b ab3 a2b ab 1 a3 a2 aab ab b a3b a2b a 1 a3 a2
a2b a2b ab b a3b a2 a 1 a3
a3b a3b a2b ab b a3 a2 a 1
Geometricgroup theory
1
aa2
a3
ba3b
a2b ab
Permutation group
〈(1 2 3 4), (1 4)(2 3)〉
Representation theory⟨(0 −11 0
),(−1 00 1
)⟩
Combinatorialgroup theory
〈a, b | a4 = b2 = 1, ba = a−1b〉
D8
symmetries of a square
ab
Abstract group× 1 a a2 a3 b ab a2b a3b1 1 a a2 a3 b ab a2b a3ba a a2 a3 1 ab a2b a3b ba2 a2 a3 1 a a2b a3b b a2ba3 a3 1 a a2 a3b b ab a2bb b ab3 a2b ab 1 a3 a2 aab ab b a3b a2b a 1 a3 a2
a2b a2b ab b a3b a2 a 1 a3
a3b a3b a2b ab b a3 a2 a 1
Geometricgroup theory
1
aa2
a3
ba3b
a2b ab
Abstract groups
A group is a set together with a binary operation satisfying . . .
D8 = 1, a, a2, a3, b, ab, a2b, a3b
× 1 a a2 a3 b ab a2b a3b1 1 a a2 a3 b ab a2b a3ba a a2 a3 1 ab a2b a3b ba2 a2 a3 1 a a2b a3b b a2ba3 a3 1 a a2 a3b b ab a2bb b ab3 a2b ab 1 a3 a2 aab ab b a3b a2b a 1 a3 a2
a2b a2b ab b a3b a2 a 1 a3
a3b a3b a2b ab b a3 a2 a 1
Abstract groups
A group is a set together with a binary operation satisfying . . .
D8 = 1, a, a2, a3, b, ab, a2b, a3b
× 1 a a2 a3 b ab a2b a3b1 1 a a2 a3 b ab a2b a3ba a a2 a3 1 ab a2b a3b ba2 a2 a3 1 a a2b a3b b a2ba3 a3 1 a a2 a3b b ab a2bb b ab3 a2b ab 1 a3 a2 aab ab b a3b a2b a 1 a3 a2
a2b a2b ab b a3b a2 a 1 a3
a3b a3b a2b ab b a3 a2 a 1
Abstract groups
A group is a set together with a binary operation satisfying . . .
D8 = 1, a, a2, a3, b, ab, a2b, a3b
× 1 a a2 a3 b ab a2b a3b1 1 a a2 a3 b ab a2b a3ba a a2 a3 1 ab a2b a3b ba2 a2 a3 1 a a2b a3b b a2ba3 a3 1 a a2 a3b b ab a2bb b ab3 a2b ab 1 a3 a2 aab ab b a3b a2b a 1 a3 a2
a2b a2b ab b a3b a2 a 1 a3
a3b a3b a2b ab b a3 a2 a 1
Permutation group
〈(1 2 3 4), (1 4)(2 3)〉
Representation theory⟨(0 −11 0
),(−1 00 1
)⟩
Combinatorialgroup theory
〈a, b | a4 = b2 = 1, ba = a−1b〉
D8
symmetries of a square
ab
Abstract group× 1 a a2 a3 b ab a2b a3b1 1 a a2 a3 b ab a2b a3ba a a2 a3 1 ab a2b a3b ba2 a2 a3 1 a a2b a3b b a2ba3 a3 1 a a2 a3b b ab a2bb b ab3 a2b ab 1 a3 a2 aab ab b a3b a2b a 1 a3 a2
a2b a2b ab b a3b a2 a 1 a3
a3b a3b a2b ab b a3 a2 a 1
Geometricgroup theory
1
aa2
a3
ba3b
a2b ab
Permutation groups
1
23
4
ab
symmetries of an oblong
1
23
4
1
2
3
4
a = (1 2 3 4) b = (1 4)(2 3)
D8 = 〈(1 2 3 4), (1 4)(2 3)〉 6 Sym(4)
id, (1 2)(3 4), (1 4)(2 3), (1 3)(2 4)
id, (1 3), (2 4), (1 3)(2 4)
A permutation group is a subgroup G 6 Sym(Ω).
Cayley’s Theorem Every group is a permutation group.
Permutation groups
1
23
4
ab
symmetries of an oblong
1
23
4
1
2
3
4
a = (1 2 3 4) b = (1 4)(2 3)
D8 = 〈(1 2 3 4), (1 4)(2 3)〉 6 Sym(4)
id, (1 2)(3 4), (1 4)(2 3), (1 3)(2 4)
id, (1 3), (2 4), (1 3)(2 4)
A permutation group is a subgroup G 6 Sym(Ω).
Cayley’s Theorem Every group is a permutation group.
Permutation groups
1
23
4
ab
symmetries of an oblong
1
23
4
1
2
3
4
a = (1 2 3 4) b = (1 4)(2 3)
D8 = 〈(1 2 3 4), (1 4)(2 3)〉 6 Sym(4)
id, (1 2)(3 4), (1 4)(2 3), (1 3)(2 4)
id, (1 3), (2 4), (1 3)(2 4)
A permutation group is a subgroup G 6 Sym(Ω).
Cayley’s Theorem Every group is a permutation group.
Permutation groups
1
23
4
ab
symmetries of an oblong
1
23
4
1
2
3
4
a = (1 2 3 4) b = (1 4)(2 3)
D8 = 〈(1 2 3 4), (1 4)(2 3)〉 6 Sym(4)
id, (1 2)(3 4), (1 4)(2 3), (1 3)(2 4)
id, (1 3), (2 4), (1 3)(2 4)
A permutation group is a subgroup G 6 Sym(Ω).
Cayley’s Theorem Every group is a permutation group.
Permutation groups
1
23
4
ab
symmetries of an oblong
1
23
4
1
2
3
4
a = (1 2 3 4) b = (1 4)(2 3)
D8 = 〈(1 2 3 4), (1 4)(2 3)〉 6 Sym(4)
id, (1 2)(3 4), (1 4)(2 3), (1 3)(2 4)
id, (1 3), (2 4), (1 3)(2 4)
A permutation group is a subgroup G 6 Sym(Ω).
Cayley’s Theorem Every group is a permutation group.
Permutation groups
1
23
4
absymmetries of an oblong
1
23
4
1
2
3
4
a = (1 2 3 4) b = (1 4)(2 3)
D8 = 〈(1 2 3 4), (1 4)(2 3)〉 6 Sym(4)
id, (1 2)(3 4), (1 4)(2 3), (1 3)(2 4)
id, (1 3), (2 4), (1 3)(2 4)
A permutation group is a subgroup G 6 Sym(Ω).
Cayley’s Theorem Every group is a permutation group.
Permutation groups
1
23
4
absymmetries of an oblong
1
23
4
1
2
3
4
a = (1 2 3 4) b = (1 4)(2 3)
D8 = 〈(1 2 3 4), (1 4)(2 3)〉 6 Sym(4)
id, (1 2)(3 4), (1 4)(2 3), (1 3)(2 4)
id, (1 3), (2 4), (1 3)(2 4)
A permutation group is a subgroup G 6 Sym(Ω).
Cayley’s Theorem Every group is a permutation group.
Permutation groups
1
23
4
absymmetries of an oblong
1
23
4
1
2
3
4
a = (1 2 3 4) b = (1 4)(2 3)
D8 = 〈(1 2 3 4), (1 4)(2 3)〉 6 Sym(4)
id, (1 2)(3 4), (1 4)(2 3), (1 3)(2 4)
id, (1 3), (2 4), (1 3)(2 4)
A permutation group is a subgroup G 6 Sym(Ω).
Cayley’s Theorem Every group is a permutation group.
Permutation groups
1
23
4
absymmetries of an oblong
1
23
41
2
3
4
a = (1 2 3 4) b = (1 4)(2 3)
D8 = 〈(1 2 3 4), (1 4)(2 3)〉 6 Sym(4)
id, (1 2)(3 4), (1 4)(2 3), (1 3)(2 4)
id, (1 3), (2 4), (1 3)(2 4)
A permutation group is a subgroup G 6 Sym(Ω).
Cayley’s Theorem Every group is a permutation group.
Permutation groups
1
23
4
absymmetries of an oblong
1
23
41
2
3
4
a = (1 2 3 4) b = (1 4)(2 3)
D8 = 〈(1 2 3 4), (1 4)(2 3)〉 6 Sym(4)
id, (1 2)(3 4), (1 4)(2 3), (1 3)(2 4)
id, (1 3), (2 4), (1 3)(2 4)
A permutation group is a subgroup G 6 Sym(Ω).
Cayley’s Theorem Every group is a permutation group.
Permutation group
〈(1 2 3 4), (1 4)(2 3)〉
Representation theory⟨(0 −11 0
),(−1 00 1
)⟩
Combinatorialgroup theory
〈a, b | a4 = b2 = 1, ba = a−1b〉
D8
symmetries of a square
ab
Abstract group× 1 a a2 a3 b ab a2b a3b1 1 a a2 a3 b ab a2b a3ba a a2 a3 1 ab a2b a3b ba2 a2 a3 1 a a2b a3b b a2ba3 a3 1 a a2 a3b b ab a2bb b ab3 a2b ab 1 a3 a2 aab ab b a3b a2b a 1 a3 a2
a2b a2b ab b a3b a2 a 1 a3
a3b a3b a2b ab b a3 a2 a 1
Geometricgroup theory
1
aa2
a3
ba3b
a2b ab
Matrix groups
ab
a =
(0 −11 0
)b =
(−1 00 1
)
D8 =
⟨(0 −11 0
),
(−1 00 1
)⟩6 GL2(R)
A representation is a homomorphism ϕ : G→ GLd(F).
Matrix groups
ab
a =
(0 −11 0
)b =
(−1 00 1
)
D8 =
⟨(0 −11 0
),
(−1 00 1
)⟩6 GL2(R)
A representation is a homomorphism ϕ : G→ GLd(F).
Matrix groups
ab
a =
(0 −11 0
)b =
(−1 00 1
)
D8 =
⟨(0 −11 0
),
(−1 00 1
)⟩6 GL2(R)
A representation is a homomorphism ϕ : G→ GLd(F).
Matrix groups
ab
a =
(0 −11 0
)b =
(−1 00 1
)
D8 =
⟨(0 −11 0
),
(−1 00 1
)⟩6 GL2(R)
A representation is a homomorphism ϕ : G→ GLd(F).
Representation theory
ab
a =
(0 −11 0
)b =
(−1 00 1
)
D8 =
⟨(0 −11 0
),
(−1 00 1
)⟩6 GL2(R)
A representation is a homomorphism ϕ : G→ GLd(F).
Theorem (Burnside, 1904)Let p and q be prime. Any group of order paqb is soluble.
William Burnside
Theorem (Burnside, 1904)Let p and q be prime. Any group of order paqb is soluble.
Ambrose Burnside
Permutation group
〈(1 2 3 4), (1 4)(2 3)〉
Representation theory⟨(0 −11 0
),(−1 00 1
)⟩
Combinatorialgroup theory
〈a, b | a4 = b2 = 1, ba = a−1b〉
D8
symmetries of a square
ab
Abstract group× 1 a a2 a3 b ab a2b a3b1 1 a a2 a3 b ab a2b a3ba a a2 a3 1 ab a2b a3b ba2 a2 a3 1 a a2b a3b b a2ba3 a3 1 a a2 a3b b ab a2bb b ab3 a2b ab 1 a3 a2 aab ab b a3b a2b a 1 a3 a2
a2b a2b ab b a3b a2 a 1 a3
a3b a3b a2b ab b a3 a2 a 1
Geometricgroup theory
1
aa2
a3
ba3b
a2b ab
Geometric group theory
1
aa2
a3
ba3b
a2b ab
Cayley graph of D8 with respect to a, b
Γ(D8, a, b)
a-2
a-1
1
a
a2a-2b
a-1bb
ab
a2b
Γ(D∞, a, b)
1
aa2
a3
ba3b
a2b ab
Γ(D8, a, b)
bxb bx b 1 x xb xbx
Γ(D∞, x, b)
−3 −2 −1 0 1 2 3
Γ(Z, 1, −1)
a-2
a-1
1
a
a2a-2b
a-1bb
ab
a2b
Γ(D∞, a, b)
1
aa2
a3
ba3b
a2b ab
Γ(D8, a, b)
bxb bx b 1 x xb xbx
Γ(D∞, x, b)
−3 −2 −1 0 1 2 3
Γ(Z, 1, −1)
a-2
a-1
1
a
a2a-2b
a-1bb
ab
a2b
Γ(D∞, a, b)
bxb bx b 1 x xb xbx
Γ(D∞, x, b)
−3 −2 −1 0 1 2 3
Γ(Z, 1, −1)
a-2
a-1
1
a
a2a-2b
a-1bb
ab
a2b
Γ(D∞, a, b)
bxbx
bx
1
xb
xbxbbxbxb
bxbb
x
xbx
Γ(D∞, a, b)
bxb bx b 1 x xb xbx
Γ(D∞, x, b)
−3 −2 −1 0 1 2 3
Γ(Z, 1, −1)
a-2
a-1
1
a
a2a-2b
a-1bb
ab
a2b
Γ(D∞, a, b)
bxbx
bx
1
xb
xbxbbxbxb
bxbb
x
xbx
Γ(D∞, , b)
bxb bx b 1 x xb xbx
Γ(D∞, x, b)
−3 −2 −1 0 1 2 3
Γ(Z, 1, −1)
a-2
a-1
1
a
a2a-2b
a-1bb
ab
a2b
Γ(D∞, a, b)
bxbx
bx
1
xb
xbxbbxbxb
bxbb
x
xbx
Γ(D∞, , b)
bxb bx b 1 x xb xbx
Γ(D∞, x, b)
−3 −2 −1 0 1 2 3
Γ(Z, 1, −1)
a-2
a-1
1
a
a2a-2b
a-1bb
ab
a2b
Γ(D∞, a, b)
bxb bx b 1 x xb xbx
Γ(D∞, x, b)
−3 −2 −1 0 1 2 3
Γ(Z, 1, −1)
a-2
a-1
1
a
a2a-2b
a-1bb
ab
a2b
Γ(D∞, a, b)
Different generating sets givedifferent Cayley graphs
– Think up to quasiisometry
Different groups have the sameCayley graph
– Get used to it
bxb bx b 1 x xb xbx
Γ(D∞, x, b)
−3 −2 −1 0 1 2 3
Γ(Z, 1, −1)
a-2
a-1
1
a
a2a-2b
a-1bb
ab
a2b
Γ(D∞, a, b)
Different generating sets givedifferent Cayley graphs
– Think up to quasiisometry
Different groups have the sameCayley graph
– Get used to it
bxb bx b 1 x xb xbx
Γ(D∞, x, b)
−3 −2 −1 0 1 2 3
Γ(Z, 1, −1)
a-2
a-1
1
a
a2a-2b
a-1bb
ab
a2b
Γ(D∞, a, b)
Different generating sets givedifferent Cayley graphs
– Think up to quasiisometry
Different groups have the sameCayley graph
– Get used to it
bxb bx b 1 x xb xbx
Γ(D∞, x, b)
−3 −2 −1 0 1 2 3
Γ(Z, 1, −1)
a-2
a-1
1
a
a2a-2b
a-1bb
ab
a2b
Γ(D∞, a, b)
Different generating sets givedifferent Cayley graphs
– Think up to quasiisometry
Different groups have the sameCayley graph
– Get used to it
bxb bx b 1 x xb xbx
Γ(D∞, x, b)
−3 −2 −1 0 1 2 3
Γ(Z, 1, −1)
a-2
a-1
1
a
a2a-2b
a-1bb
ab
a2b
Γ(D∞, a, b)
Different generating sets givedifferent Cayley graphs
– Think up to quasiisometry
Different groups have the sameCayley graph
– Get used to it
bxb bx b 1 x xb xbx
Γ(D∞, x, b)
−3 −2 −1 0 1 2 3
Γ(Z, 1, −1)
“If H is a finite index subgroup of G, then H looks the same as G.”– a geometric group theorist
The index of a subgroup H 6 G is the cardinality of the set Hg | g ∈ G.
〈a〉 is a finite index subgroup of D∞ = 〈a, b〉
If H is a finite index subgroup of G, then Γ(H) is quasiisometric to Γ(G).
a-2
a-1
1
a
a2a-2b
a-1bb
ab
a2b
is quasiisometric to −3 −2 −1 0 1 2 3
A group is virtually purple if it has finite index subgroup that is purple.
〈a〉 is cyclic so D∞ is virtually cyclic
“If H is a finite index subgroup of G, then H looks the same as G.”– a geometric group theorist
The index of a subgroup H 6 G is the cardinality of the set Hg | g ∈ G.
〈a〉 is a finite index subgroup of D∞ = 〈a, b〉
If H is a finite index subgroup of G, then Γ(H) is quasiisometric to Γ(G).
a-2
a-1
1
a
a2a-2b
a-1bb
ab
a2b
is quasiisometric to −3 −2 −1 0 1 2 3
A group is virtually purple if it has finite index subgroup that is purple.
〈a〉 is cyclic so D∞ is virtually cyclic
“If H is a finite index subgroup of G, then H looks the same as G.”– a geometric group theorist
The index of a subgroup H 6 G is the cardinality of the set Hg | g ∈ G.
〈a〉 is a finite index subgroup of D∞ = 〈a, b〉
If H is a finite index subgroup of G, then Γ(H) is quasiisometric to Γ(G).
a-2
a-1
1
a
a2a-2b
a-1bb
ab
a2b
is quasiisometric to −3 −2 −1 0 1 2 3
A group is virtually purple if it has finite index subgroup that is purple.
〈a〉 is cyclic so D∞ is virtually cyclic
“If H is a finite index subgroup of G, then H looks the same as G.”– a geometric group theorist
The index of a subgroup H 6 G is the cardinality of the set Hg | g ∈ G.
〈a〉 is a finite index subgroup of D∞ = 〈a, b〉
If H is a finite index subgroup of G, then Γ(H) is quasiisometric to Γ(G).
a-2
a-1
1
a
a2a-2b
a-1bb
ab
a2b
is quasiisometric to −3 −2 −1 0 1 2 3
A group is virtually purple if it has finite index subgroup that is purple.
〈a〉 is cyclic so D∞ is virtually cyclic
“If H is a finite index subgroup of G, then H looks the same as G.”– a geometric group theorist
The index of a subgroup H 6 G is the cardinality of the set Hg | g ∈ G.
〈a〉 is a finite index subgroup of D∞ = 〈a, b〉
If H is a finite index subgroup of G, then Γ(H) is quasiisometric to Γ(G).
a-2
a-1
1
a
a2a-2b
a-1bb
ab
a2b
is quasiisometric to −3 −2 −1 0 1 2 3
A group is virtually purple if it has finite index subgroup that is purple.
〈a〉 is cyclic so D∞ is virtually cyclic
“If H is a finite index subgroup of G, then H looks the same as G.”– a geometric group theorist
The index of a subgroup H 6 G is the cardinality of the set Hg | g ∈ G.
〈a〉 is a finite index subgroup of D∞ = 〈a, b〉
If H is a finite index subgroup of G, then Γ(H) is quasiisometric to Γ(G).
a-2
a-1
1
a
a2a-2b
a-1bb
ab
a2b
is quasiisometric to −3 −2 −1 0 1 2 3
A group is virtually purple if it has finite index subgroup that is purple.
〈a〉 is cyclic so D∞ is virtually cyclic
“If H is a finite index subgroup of G, then H looks the same as G.”– a geometric group theorist
The index of a subgroup H 6 G is the cardinality of the set Hg | g ∈ G.
〈a〉 is a finite index subgroup of D∞ = 〈a, b〉
If H is a finite index subgroup of G, then Γ(H) is quasiisometric to Γ(G).
a-2
a-1
1
a
a2a-2b
a-1bb
ab
a2b
is quasiisometric to −3 −2 −1 0 1 2 3
A group is virtually purple if it has finite index subgroup that is purple.
〈a〉 is cyclic so D∞ is virtually cyclic
Permutation group
〈(1 2 3 4), (1 4)(2 3)〉
Representation theory⟨(0 −11 0
),(−1 00 1
)⟩
Combinatorialgroup theory
〈a, b | a4 = b2 = 1, ba = a−1b〉
D8
symmetries of a square
ab
Abstract group× 1 a a2 a3 b ab a2b a3b1 1 a a2 a3 b ab a2b a3ba a a2 a3 1 ab a2b a3b ba2 a2 a3 1 a a2b a3b b a2ba3 a3 1 a a2 a3b b ab a2bb b ab3 a2b ab 1 a3 a2 aab ab b a3b a2b a 1 a3 a2
a2b a2b ab b a3b a2 a 1 a3
a3b a3b a2b ab b a3 a2 a 1
Geometricgroup theory
1
aa2
a3
ba3b
a2b ab
Combinatorial group theory
D8 = 〈a, b | a4 = 1, b2 = 1, ba = a−1b〉
a presentation for D8
Examples
〈a | a4 = 1〉
= C4 ∼= Z/4Z
〈a | 〉
∼= Z ∼= F1
〈a, b | b2 = 1, ba = a−1b〉
= D∞
〈a, b | 〉
= F2
A group is free if it admits a presentation of the form 〈A | 〉.
Combinatorial group theory
D8 = 〈a, b | a4 = 1, b2 = 1, ba = a−1b〉
a presentation for D8
Examples
〈a | a4 = 1〉
= C4 ∼= Z/4Z
〈a | 〉
∼= Z ∼= F1
〈a, b | b2 = 1, ba = a−1b〉
= D∞
〈a, b | 〉
= F2
A group is free if it admits a presentation of the form 〈A | 〉.
Combinatorial group theory
D8 = 〈a, b | a4 = 1, b2 = 1, ba = a−1b〉
a presentation for D8
Examples
〈a | a4 = 1〉
= C4 ∼= Z/4Z
〈a | 〉
∼= Z ∼= F1
〈a, b | b2 = 1, ba = a−1b〉
= D∞
〈a, b | 〉
= F2
A group is free if it admits a presentation of the form 〈A | 〉.
Combinatorial group theory
D8 = 〈a, b | a4 = 1, b2 = 1, ba = a−1b〉
a presentation for D8
Examples
〈a | a4 = 1〉
= C4 ∼= Z/4Z
〈a | 〉
∼= Z ∼= F1
〈a, b | b2 = 1, ba = a−1b〉
= D∞
〈a, b | 〉
= F2
A group is free if it admits a presentation of the form 〈A | 〉.
Combinatorial group theory
D8 = 〈a, b | a4 = 1, b2 = 1, ba = a−1b〉
a presentation for D8
Examples
〈a | a4 = 1〉 = C4 ∼= Z/4Z
〈a | 〉
∼= Z ∼= F1
〈a, b | b2 = 1, ba = a−1b〉
= D∞
〈a, b | 〉
= F2
A group is free if it admits a presentation of the form 〈A | 〉.
Combinatorial group theory
D8 = 〈a, b | a4 = 1, b2 = 1, ba = a−1b〉
a presentation for D8
Examples
〈a | a4 = 1〉 = C4 ∼= Z/4Z
〈a | 〉
∼= Z ∼= F1
〈a, b | b2 = 1, ba = a−1b〉
= D∞
〈a, b | 〉
= F2
A group is free if it admits a presentation of the form 〈A | 〉.
Combinatorial group theory
D8 = 〈a, b | a4 = 1, b2 = 1, ba = a−1b〉
a presentation for D8
Examples
〈a | a4 = 1〉 = C4 ∼= Z/4Z
〈a | 〉 ∼= Z
∼= F1
〈a, b | b2 = 1, ba = a−1b〉
= D∞
〈a, b | 〉
= F2
A group is free if it admits a presentation of the form 〈A | 〉.
Combinatorial group theory
D8 = 〈a, b | a4 = 1, b2 = 1, ba = a−1b〉
a presentation for D8
Examples
〈a | a4 = 1〉 = C4 ∼= Z/4Z
〈a | 〉 ∼= Z
∼= F1
〈a, b | b2 = 1, ba = a−1b〉
= D∞
〈a, b | 〉
= F2
A group is free if it admits a presentation of the form 〈A | 〉.
Combinatorial group theory
D8 = 〈a, b | a4 = 1, b2 = 1, ba = a−1b〉
a presentation for D8
Examples
〈a | a4 = 1〉 = C4 ∼= Z/4Z
〈a | 〉 ∼= Z
∼= F1
〈a, b | b2 = 1, ba = a−1b〉 = D∞
〈a, b | 〉
= F2
A group is free if it admits a presentation of the form 〈A | 〉.
Combinatorial group theory
D8 = 〈a, b | a4 = 1, b2 = 1, ba = a−1b〉
a presentation for D8
Examples
〈a | a4 = 1〉 = C4 ∼= Z/4Z
〈a | 〉 ∼= Z
∼= F1
〈a, b | b2 = 1, ba = a−1b〉 = D∞
〈a, b | 〉
= F2
A group is free if it admits a presentation of the form 〈A | 〉.
Combinatorial group theory
D8 = 〈a, b | a4 = 1, b2 = 1, ba = a−1b〉
a presentation for D8
Examples
〈a | a4 = 1〉 = C4 ∼= Z/4Z
〈a | 〉 ∼= Z
∼= F1
〈a, b | b2 = 1, ba = a−1b〉 = D∞
〈a, b | 〉
= F2
A group is free if it admits a presentation of the form 〈A | 〉.
Combinatorial group theory
D8 = 〈a, b | a4 = 1, b2 = 1, ba = a−1b〉
a presentation for D8
Examples
〈a | a4 = 1〉 = C4 ∼= Z/4Z
〈a | 〉 ∼= Z
∼= F1
〈a, b | b2 = 1, ba = a−1b〉 = D∞
〈a, b | 〉 = F2
A group is free if it admits a presentation of the form 〈A | 〉.
Combinatorial group theory
D8 = 〈a, b | a4 = 1, b2 = 1, ba = a−1b〉
a presentation for D8
Examples
〈a | a4 = 1〉 = C4 ∼= Z/4Z
〈a | 〉 ∼= Z ∼= F1
〈a, b | b2 = 1, ba = a−1b〉 = D∞
〈a, b | 〉 = F2
A group is free if it admits a presentation of the form 〈A | 〉.
Permutation group
〈(1 2 3 4), (1 4)(2 3)〉
Representation theory⟨(0 −11 0
),(−1 00 1
)⟩
Combinatorialgroup theory
〈a, b | a4 = b2 = 1, ba = a−1b〉
D8
symmetries of a square
ab
Abstract group× 1 a a2 a3 b ab a2b a3b1 1 a a2 a3 b ab a2b a3ba a a2 a3 1 ab a2b a3b ba2 a2 a3 1 a a2b a3b b a2ba3 a3 1 a a2 a3b b ab a2bb b ab3 a2b ab 1 a3 a2 aab ab b a3b a2b a 1 a3 a2
a2b a2b ab b a3b a2 a 1 a3
a3b a3b a2b ab b a3 a2 a 1
Geometricgroup theory
1
aa2
a3
ba3b
a2b ab
The problem with presentations
Tell me something about this group
〈a, b, c | ab = b2a, bc = c2b, ca = a2c〉
It’s trivial.
The sad realityDetermining whether a presentation gives the trivial group is undecidable.
The Word Problem (Dehn, 1911)Fix a finitely generated group G = 〈A〉. Determine whether a word in A istrivial in G.
Theorem (Novikov, 1954)There are groups for which the Word Problem is undecidable.
The problem with presentations
Tell me something about this group
〈a, b, c | ab = b2a, bc = c2b, ca = a2c〉
It’s trivial.
The sad realityDetermining whether a presentation gives the trivial group is undecidable.
The Word Problem (Dehn, 1911)Fix a finitely generated group G = 〈A〉. Determine whether a word in A istrivial in G.
Theorem (Novikov, 1954)There are groups for which the Word Problem is undecidable.
The problem with presentations
Tell me something about this group
〈a, b, c | ab = b2a, bc = c2b, ca = a2c〉
It’s trivial.
The sad realityDetermining whether a presentation gives the trivial group is undecidable.
The Word Problem (Dehn, 1911)Fix a finitely generated group G = 〈A〉. Determine whether a word in A istrivial in G.
Theorem (Novikov, 1954)There are groups for which the Word Problem is undecidable.
The problem with presentations
Tell me something about this group
〈a, b, c | ab = b2a, bc = c2b, ca = a2c〉
It’s trivial.
The sad realityDetermining whether a presentation gives the trivial group is undecidable.
The Word Problem (Dehn, 1911)Fix a finitely generated group G = 〈A〉. Determine whether a word in A istrivial in G.
Theorem (Novikov, 1954)There are groups for which the Word Problem is undecidable.
The problem with presentations
Tell me something about this group
〈a, b, c | ab = b2a, bc = c2b, ca = a2c〉
It’s trivial.
The sad realityDetermining whether a presentation gives the trivial group is undecidable.
The Word Problem (Dehn, 1911)Fix a finitely generated group G = 〈A〉. Determine whether a word in A istrivial in G.
Theorem (Novikov, 1954)There are groups for which the Word Problem is undecidable.
Theorem
(Anisimov, 1971)
A group G is finite =⇒ word problem for G is solvable
by an automaton
1
aa2
a3
ba3b
a2b ab
1
D8 = 〈a, b | a4 = b2 = 1, ba = ab3〉
abaaabbaab
Theorem
(Anisimov, 1971)
A group G is finite =⇒ word problem for G is solvable
by an automaton
1
aa2
a3
ba3b
a2b ab
1
D8 = 〈a, b | a4 = b2 = 1, ba = ab3〉
abaaabbaab
Theorem
(Anisimov, 1971)
A group G is finite =⇒ word problem for G is solvable
by an automaton
1
aa2
a3
ba3b
a2b ab
1
D8 = 〈a, b | a4 = b2 = 1, ba = ab3〉
abaaabbaab
Theorem
(Anisimov, 1971)
A group G is finite =⇒ word problem for G is solvable
by an automaton
1
aa2
a3
ba3b
a2b ab
1
D8 = 〈a, b | a4 = b2 = 1, ba = ab3〉
abaaabbaab
Theorem
(Anisimov, 1971)
A group G is finite =⇒ word problem for G is solvable
by an automaton
1
aa2
a3
ba3b
a2b ab
1
D8 = 〈a, b | a4 = b2 = 1, ba = ab3〉 abaaabbaab
Theorem
(Anisimov, 1971)
A group G is finite =⇒ word problem for G is solvable
by an automaton
1
aa2
a3
ba3b
a2b ab
11
D8 = 〈a, b | a4 = b2 = 1, ba = ab3〉 abaaabbaab
Theorem
(Anisimov, 1971)
A group G is finite =⇒ word problem for G is solvable
by an automaton
1
aa2
a3
ba3b
a2b ab
1
a
D8 = 〈a, b | a4 = b2 = 1, ba = ab3〉 abaaabbaab
Theorem
(Anisimov, 1971)
A group G is finite =⇒ word problem for G is solvable
by an automaton
1
aa2
a3
ba3b
a2b ab
1
ab
D8 = 〈a, b | a4 = b2 = 1, ba = ab3〉 abaaabbaab
Theorem
(Anisimov, 1971)
A group G is finite =⇒ word problem for G is solvable
by an automaton
1
aa2
a3
ba3b
a2b ab
1
b
D8 = 〈a, b | a4 = b2 = 1, ba = ab3〉 abaaabbaab
Theorem
(Anisimov, 1971)
A group G is finite =⇒ word problem for G is solvable
by an automaton
1
aa2
a3
ba3b
a2b ab
1
a3b
D8 = 〈a, b | a4 = b2 = 1, ba = ab3〉 abaaabbaab
Theorem
(Anisimov, 1971)
A group G is finite =⇒ word problem for G is solvable
by an automaton
1
aa2
a3
ba3b
a2b ab
1
a2b
D8 = 〈a, b | a4 = b2 = 1, ba = ab3〉 abaaabbaab
Theorem
(Anisimov, 1971)
A group G is finite =⇒ word problem for G is solvable
by an automaton
1
aa2
a3
ba3b
a2b ab
1
a2
D8 = 〈a, b | a4 = b2 = 1, ba = ab3〉 abaaabbaab
Theorem
(Anisimov, 1971)
A group G is finite =⇒ word problem for G is solvable
by an automaton
1
aa2
a3
ba3b
a2b ab
1
a2b
D8 = 〈a, b | a4 = b2 = 1, ba = ab3〉 abaaabbaab
Theorem
(Anisimov, 1971)
A group G is finite =⇒ word problem for G is solvable
by an automaton
1
aa2
a3
ba3b
a2b ab
1
ab
D8 = 〈a, b | a4 = b2 = 1, ba = ab3〉 abaaabbaab
Theorem
(Anisimov, 1971)
A group G is finite =⇒ word problem for G is solvable
by an automaton
1
aa2
a3
ba3b
a2b ab
1
b
D8 = 〈a, b | a4 = b2 = 1, ba = ab3〉 abaaabbaab
Theorem
(Anisimov, 1971)
A group G is finite =⇒ word problem for G is solvable
by an automaton
1
aa2
a3
ba3b
a2b ab
11
D8 = 〈a, b | a4 = b2 = 1, ba = ab3〉 abaaabbaab
Theorem
(Anisimov, 1971)
A group G is finite =⇒ word problem for G is solvable
by an automaton
1
aa2
a3
ba3b
a2b ab
11
D8 = 〈a, b | a4 = b2 = 1, ba = ab3〉 abaaabbaab
Theorem
(Anisimov, 1971)
A group G is finite =⇒ word problem for G is solvable by an automaton
1
aa2
a3
ba3b
a2b ab
1
D8 = 〈a, b | a4 = b2 = 1, ba = ab3〉 abaaabbaab
Theorem
(Anisimov, 1971)
A group G is finite ⇐⇒ word problem for G is solvable by an automaton
1
aa2
a3
ba3b
a2b ab
1
D8 = 〈a, b | a4 = b2 = 1, ba = ab3〉 abaaabbaab
Theorem (Anisimov, 1971)A group G is finite ⇐⇒ word problem for G is solvable by an automaton
1
aa2
a3
ba3b
a2b ab
1
D8 = 〈a, b | a4 = b2 = 1, ba = ab3〉 abaaabbaab
TheoremA group G is free =⇒ word problem for G is solvable
by a stack
F2 = 〈a, b | 〉 ab-1a-1abaa-1a-1
a
b-1
a-1
a
Theorem
(Muller–Schupp, 1983)
A group G is virtually free word problem for G is solvableby a push down automaton
TheoremA group G is free =⇒ word problem for G is solvable by a stack
F2 = 〈a, b | 〉 ab-1a-1abaa-1a-1
a
b-1
a-1
a
Theorem
(Muller–Schupp, 1983)
A group G is virtually free word problem for G is solvableby a push down automaton
TheoremA group G is free =⇒ word problem for G is solvable by a stack
F2 = 〈a, b | 〉 ab-1a-1abaa-1a-1
a
b-1
a-1
a
Theorem
(Muller–Schupp, 1983)
A group G is virtually free word problem for G is solvableby a push down automaton
TheoremA group G is free =⇒ word problem for G is solvable by a stack
F2 = 〈a, b | 〉 ab-1a-1abaa-1a-1
a
b-1
a-1
a
Theorem
(Muller–Schupp, 1983)
A group G is virtually free word problem for G is solvableby a push down automaton
TheoremA group G is free =⇒ word problem for G is solvable by a stack
F2 = 〈a, b | 〉
ab-1a-1abaa-1a-1
a
b-1
a-1
a
Theorem
(Muller–Schupp, 1983)
A group G is virtually free word problem for G is solvableby a push down automaton
TheoremA group G is free =⇒ word problem for G is solvable by a stack
F2 = 〈a, b | 〉 ab-1a-1abaa-1a-1
a
b-1
a-1
a
Theorem
(Muller–Schupp, 1983)
A group G is virtually free word problem for G is solvableby a push down automaton
TheoremA group G is free =⇒ word problem for G is solvable by a stack
F2 = 〈a, b | 〉 ab-1a-1abaa-1a-1
a
b-1
a-1
a
Theorem
(Muller–Schupp, 1983)
A group G is virtually free word problem for G is solvableby a push down automaton
TheoremA group G is free =⇒ word problem for G is solvable by a stack
F2 = 〈a, b | 〉 ab-1a-1abaa-1a-1
a
b-1
a-1
a
Theorem
(Muller–Schupp, 1983)
A group G is virtually free word problem for G is solvableby a push down automaton
TheoremA group G is free =⇒ word problem for G is solvable by a stack
F2 = 〈a, b | 〉 ab-1a-1abaa-1a-1
a
b-1
a-1
a
Theorem
(Muller–Schupp, 1983)
A group G is virtually free word problem for G is solvableby a push down automaton
TheoremA group G is free =⇒ word problem for G is solvable by a stack
F2 = 〈a, b | 〉 ab-1a-1abaa-1a-1
a
b-1
a-1
a
Theorem
(Muller–Schupp, 1983)
A group G is virtually free word problem for G is solvableby a push down automaton
TheoremA group G is free =⇒ word problem for G is solvable by a stack
F2 = 〈a, b | 〉 ab-1a-1abaa-1a-1
a
b-1
a-1
a
Theorem
(Muller–Schupp, 1983)
A group G is virtually free word problem for G is solvableby a push down automaton
TheoremA group G is free =⇒ word problem for G is solvable by a stack
F2 = 〈a, b | 〉 ab-1a-1abaa-1a-1
a
b-1
a-1
a
Theorem
(Muller–Schupp, 1983)
A group G is virtually free word problem for G is solvableby a push down automaton
TheoremA group G is free =⇒ word problem for G is solvable by a stack
F2 = 〈a, b | 〉 ab-1a-1abaa-1a-1
a
b-1
a-1
a
Theorem
(Muller–Schupp, 1983)
A group G is virtually free word problem for G is solvableby a push down automaton
TheoremA group G is free =⇒ word problem for G is solvable by a stack
F2 = 〈a, b | 〉 ab-1a-1abaa-1a-1
a
b-1
a-1
a
Theorem
(Muller–Schupp, 1983)
A group G is virtually free word problem for G is solvableby a push down automaton
TheoremA group G is free =⇒ word problem for G is solvable by a stack
F2 = 〈a, b | 〉 ab-1a-1abaa-1a-1
a
b-1
a-1
a
Theorem
(Muller–Schupp, 1983)
A group G is virtually free word problem for G is solvableby a push down automaton
TheoremA group G is free =⇒ word problem for G is solvable by a stack
F2 = 〈a, b | 〉 ab-1a-1abaa-1a-1
a
b-1
a-1
a
Theorem
(Muller–Schupp, 1983)
A group G is virtually free word problem for G is solvableby a push down automaton
TheoremA group G is free =⇒ word problem for G is solvable by a stack
F2 = 〈a, b | 〉 ab-1a-1abaa-1a-1
a
b-1
a-1
a
Theorem
(Muller–Schupp, 1983)
A group G is virtually free word problem for G is solvableby a push down automaton
TheoremA group G is free =⇒ word problem for G is solvable by a stack
F2 = 〈a, b | 〉 ab-1a-1abaa-1a-1
a
b-1
a-1
a
Theorem
(Muller–Schupp, 1983)
A group G is virtually free word problem for G is solvableby a push down automaton
TheoremA group G is free =⇒ word problem for G is solvable by a stack
F2 = 〈a, b | 〉 ab-1a-1abaa-1a-1
a
b-1
a-1
a
Theorem
(Muller–Schupp, 1983)
A group G is virtually free word problem for G is solvableby a push down automaton
TheoremA group G is free =⇒ word problem for G is solvable by a stack
F2 = 〈a, b | 〉 ab-1a-1abaa-1a-1
a
b-1
a-1
a
Theorem
(Muller–Schupp, 1983)
A group G is virtually free word problem for G is solvableby a push down automaton
TheoremA group G is free =⇒ word problem for G is solvable by a stack
F2 = 〈a, b | 〉 ab-1a-1abaa-1a-1
a
b-1
a-1
a
Theorem
(Muller–Schupp, 1983)
A group G is virtually free =⇒ word problem for G is solvableby a push down automaton
And so our scene must to the battle fly,
Where – O for pity! – we shall much disgrace
With four or five most vile and ragged foils,
Right ill-disposed in brawl ridiculous,
The name of Agincourt. Yet sit and see,
Minding true things by what their mockeries be.
– Henry V (Act IV, prologue)
TheoremA group G is free =⇒ word problem for G is solvable by a stack
F2 = 〈a, b | 〉 ab-1a-1abaa-1a-1
a
b-1
a-1
a
Theorem
(Muller–Schupp, 1983)
A group G is virtually free =⇒ word problem for G is solvableby a push down automaton
TheoremA group G is free =⇒ word problem for G is solvable by a stack
F2 = 〈a, b | 〉 ab-1a-1abaa-1a-1
a
b-1
a-1
a
Theorem
(Muller–Schupp, 1983)
A group G is virtually free ⇐⇒ word problem for G is solvableby a push down automaton
TheoremA group G is free =⇒ word problem for G is solvable by a stack
F2 = 〈a, b | 〉 ab-1a-1abaa-1a-1
a
b-1
a-1
a
Theorem (Muller–Schupp, 1983)A group G is virtually free ⇐⇒ word problem for G is solvable
by a push down automaton
Permutation group
〈(1 2 3 4), (1 4)(2 3)〉
Representation theory⟨(0 −11 0
),(−1 00 1
)⟩
Combinatorialgroup theory
〈a, b | a4 = b2 = 1, ba = a−1b〉
D8
symmetries of a square
ab
Abstract group× 1 a a2 a3 b ab a2b a3b1 1 a a2 a3 b ab a2b a3ba a a2 a3 1 ab a2b a3b ba2 a2 a3 1 a a2b a3b b a2ba3 a3 1 a a2 a3b b ab a2bb b ab3 a2b ab 1 a3 a2 aab ab b a3b a2b a 1 a3 a2
a2b a2b ab b a3b a2 a 1 a3
a3b a3b a2b ab b a3 a2 a 1
Geometricgroup theory
1
aa2
a3
ba3b
a2b ab
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