Chapter 1: Fundamental concepts of infinite group theory

• Section A: Finite presentations
  • Free groups, generators, relators, relations
  • Homomorphism Building Theorems (Universal properties)
  • Tietze transformations
  • Normal Forms
  • Presentation and Cayley complexes associated to presentations
• Section B: A catalog of finitely presented groups
  • Free, free abelian, and graph groups
  • Fundamental groups of surfaces
  • Reflection and Coxeter groups
  • Braid and Artin groups
  • Matrix groups, including the Heisenberg group
  • Baumslag-Solitar groups
• Section C: Group and homomorphism constructions
  • Automorphism groups:
    • Inner and outer automorphism groups
    • Examples
  • Extensions:
    • Definition, example: direct products
    • Thm: An extension of a finitely presented group by another finitely presented group is finitely presented.
    • Semidirect products in 3 views
    • Thm: G is a semidirect product N ⋊ H iff G is a split extension of N by H.
  • Other constructions:
    • Amalgamated products: Definition and examples
    • Thm: If A and B are finitely presented, then G:=A*_CB is finitely presented iff C is finitely generated.
    • HNN extensions: Definition and examples
    • Thm: If A is finitely presented, then G:=A*_C is finitely presented iff C is finitely generated.
  • Graph products: Definition and examples
  • Thm: If the (vertex) groups G_v are all finitely presented, then so is their graph product group.

Chapter 2: Poly properties

• Section A: Definitions, closure properties, and examples
  • Poly-P groups and subnormal series
  • Thm: Polyfinite = finite
  • Polyfree: Braid (and Artin?) groups
  • Polycyclic examples: free abelian, Heisenberg group
• Section B: Measuring "abelian-ness"
  • Solvable = polyabelian and metabelian groups, derivation length dl(G)
  • Thm: The class of solvable groups is closed with respect to subgroups, images, and extensions
  • Derived series; Thm: G is solvable iff G has an abelian normal series iff G⁽ⁿ⁾ = 1 for some n.
    Moreover, dl(G) = min { n | G⁽ⁿ⁾ = 1 }.
  • Commutator calculus and Witt-Hall identites
  • Nilpotent = polycentral-normal groups, nilpotence class c(G)
  • Thm: Abelian implies nilpotent
  • Thm: The Heisenberg group is nilpotent with class c=2 but not abelian.
  • Lower and upper central series
  • Thm: G is nilpotent iff G has an abelian central normal series iff γ_n+1 = 1 for some n iff ζ_n = G for some n.
    Moreover c(G) = min { n | γ_n+1 = 1 } = min { n | ζ_n = G }.
  • Cor: Nilpotent implies solvable
  • Thm: The Baumslag-Solitar group BS(1,2) is solvable (and metabelian) but not nilpotent
  • Thm: The class of nilpotent groups is closed with respect to subgroups, images, and finite direct products
• Section C: Finitely generated poly-P groups
  • Max property, definition and examples
  • Thm: The class of max groups is closed with respect to subgroups, images, and extensions
  • Thm: Polycyclic = solvable + max = finitely generated + solvable + max
  • Lemma: G has max iff there is no infinite strictly ascending chain of subgroups of G.
  • Thm: The Baumslag-Solitar group BS(1,2) is finitely generated and solvable but not polycyclic.
  • Thm: Finitely generated nilpotent groups are polycyclic.
  • Cor: Finitely generated nilpotent groups have max.
  • Thm (Ascending HNN extensions of Z²): Let φ : Z² → Z² be a homomorphism and let G := < a, b, t | ab = ba, tat^-1 = φ(a), tbt^-1 = φ(b) >. Then: (1) G is solvable. (2) G is polycyclic if and only if det φ = ± 1. (3) G is nilpotent-by-finite if and only if every eigenvalue of φ has absolute value 1.
  • Cor: There is a polycyclic group that is not nilpotent.
  • Isolator subgroup: Definition and examples
  • Thm (Ascending HNN extensions of finitely generated torsion-free nilpotent groups): Let N be a finitely generated torsion-free nilpotent group, and let I be the isolator of the commutator subgroup [N,N] in N. Let θ : N → N be a homomorphism, and let φ : N/I → N/I be the homomorphism induced by θ. Let G := < N, t | tnt^-1 = &theta(n) > be the ascending HNN extension of N via θ. Then: (1) G is solvable. (2) G is polycyclic if and only if det φ = ± 1. (3) G is nilpotent-by-finite if and only if every eigenvalue of φ has absolute value 1.
  • Connections to Geometrization for 3-manifolds
  • Thm: Every finitely generated torsion solvable group is finite.
  • Thm: Let P be an abstract property of groups (i.e. a property of groups invariant under isomorphism, not a property of subgroups). Then:
    • P is preserved when taking extensions if and only if poly-P = P.
    • If P is inherited by subgroups, then every subgroup of a poly-P group is poly-P.
    • If P is closed under taking quotients, then every quotient of a poly-P group is poly-P.
  • Cor: Polytrivial = trivial, polyfinite = finite, polycyclic ≠ cyclic, polyfree ≠ free, polyabelian ≠ abelian, polynilpotent ≠ nilpotent, polypolycyclic = polycyclic, polysolvable = solvable, polymax = max.
• Section D: Other group constructions and interactions with poly constructions
  • Virtually P
    • Thm: Virtually trivial = virtually finite = finite
    • Thm: Virtually cyclic ≠ cyclic, virtually free ≠ free, virtually abelian ≠ abelian
    • Thm: Virtually solvable ≠ solvable ≠ poly-virtually-abelian
  • Residually P
    • Thm: Residually trivial = trivial
    • Thm: Residually finite ≠ finite, residually cyclic ≠ cyclic, residually free ≠ free
    • Thm: Residually max ≠ max
    • Thm: If G is residually P, then G is isomorphic to a subgroup of a direct product of P groups.
    • Residually abelian = abelian
    • Thm: If G is residually P and P is a property inherited by subgroups, then every subgroup of G is residually P.
    • Thm: Polycyclic ⇒ residually finite, but finitely presented solvable =/⇒ residually finite, and finitely generated residually finite =/⇒ virtually solvable.
  • Locally P
    • Thm: Locally trivial = trivial
    • Thm: Locally finite ≠ finite, locally max ≠ max
    • Thm: Locally cyclic ≠ cyclic, locally free ≠ free
    • Thm: Locally abelian = abelian, locally (nilpotent of class c) = nilpotent of class c, and locally (solvable of derivation length d) = solvable of derivation length d, but locally nilpotent ≠ nilpotent and locally solvable ≠ solvable
    • Thm: If G is a locally P group and P is a property inherited by quotient groups, then every quotient of G is locally P.

Chapter 3: Geometric group theory

• Section A: Metric spaces
  • The word metric and path metric, and examples
  • Quasi-isometric embedding and quasi-isometry
  • Thm: Every quasi-isometry has a quasi-inverse
  • Thm: If G is a group with finite generating sets A and B, then (1) (G,d_A) is qi to (G,d_B), and (2) (G,d_A) is qi to (Γ(G,A),d_path).
  • Commensurability
  • Thm: For finitely generated groups, isometry implies commensurability implies quasi-isometry. Both converses are false.
  • Cor: If G is finitely generated and H is a finite index subgroup in G, then H is finitely generated and G is qi to H.
• Section B: Growth functions and series for finitely generated groups
  • Spherical/strict growth function &\sigma;, (ball/cumulative) growth function β, and their generating functions/growth series f_σ(z) and f_β(z)
  • Examples of growth functions and rational growth series
  • Thm: f_σ(z) = (1-z) f_β(z).
  • Lemma: If G is infinite, then β is a strictly increasing function.
  • Lipschitz equivalence of functions
  • Thm: If (G,A) and (H,B) are quasi-isometric finitely generated groups, then their (ball) growth functions are Lipschitz equivalent.
  • Thm: (Stoll) Rationality of the growth series depends upon the generating set of the group.
  • Thm: Let (G,A) and (H,B) be groups with finite generating sets and spherical growth functions f_σG(z) and f_σH(z). Then the spherical growth series f_{σG × H}(z) for the direct product G × H with respect to the generating set A ∪ B satisfies f_{σG × H}(z) = f_σG(z) f_σH(z).
  • Ex: The free abelian group Zⁿ has (cumulative) growth series (with respect to a basis generating set) f_β(z) = (1 + z)ⁿ / (1 - z)ⁿ⁺¹.
  • Thm: Let (G,A) and (H,B) be groups with finite generating sets and spherical growth functions f_σG(z) and f_σH(z). Then the spherical growth series f_{σG * H}(z) for the free product G * H with respect to the generating set A ∪ B satisfies 1 / f_{σG * H}(z) - 1 = [1 / f_σG(z) - 1] + [1 / f_σH(z) - 1].
  • Ex: The free group F_n has (cumulative) growth series (with respect to a basis generating set) f_β(z) = (1 + z) / (1 - (2n -1) z).
  • Thm: The growth function for any finitely generated group is at most exponential.
  • Thm: Suppose that the spherical growth function σ of the group G with respect to the finite generating set A is σ(n) = k (k - 1)^n-1 where k is a constant. Then G ≅ F_d * (*^e Z₂) for some d,e in N₀ with k = d + e.
  • Thm: If G is a finitely presented group, then G has solvable word problem if and only if the growth function for G is computable.
  • Results for finitely generated poly-P groups
    • Thm (Gromov): A f.g. group has polynomial growth iff it is virtually (f.g.) nilpotent
    • Thm (Milnor): A f.g. solvable group has exponential growth iff it is not virtually nilpotent
    • Thm (Grigorchuk): There exist groups of intermediate growth.
• Section C: Languages and growth
  • Regular languages:
    • Regular expressions
    • Finite state automata
    • Examples
    • Growth functions and series for languages
    • Thm: The growth series for a regular language is rational.
  • Rewriting systems:
    • Complete rewriting systems: Definitions and examples
    • Computing growth series for groups using shortlex regular complete rewriting systems
• Section D: Other measures of growth for finitely generated groups
  • Geodesic languages, strict and cumulative growth functions, and series
    • Definitions and examples
    • Prop:A graph product of groups has a weightlex complete rewriting system that is compatible with shortlex complete rewriting systems on each of the vertex groups.
    • Thm: Suppose that G = GΛ is a graph product of groups G₁,...,G_n, each G_i = < X_i > with X_i finite and X_i = X_i^-1. Let <_i be the shortlex ordering on X_i^* induced by a total orderin on X_i for each i.
      Let X := ∪_i=1ⁿ X_i, and
      letT := {t \subseteq X | t ≠ ∅; &forall i, |t &cap X_i| ≤ 1; and &forall i,j, |t &cap X_i| = 1 = |t &cap X_j| ⇒ vertices v_i, v_j adjacent in Λ}.
      Then:
      • If SL(G_i,X_i,<_i) is a regular language for all i, then G has regular geodesic languages of normal forms over the generating sets T and X.
      • If Geo(G_i,X_i) is a regular language for all i, then so is Geo(G,X).
    • Cor: Every right-angled Artin group and right-angled Coxeter group has rational growth series and rational geodesic growth series.
  • Conjugacy languages, strict and cumulative growth functions, and series
    • Definition and examples
    • Thm: If G and H are finite groups, then the conjugacy growth series for the free product G*H is rational if and only if G = Z₂ = H.
    • Cor: A right-angled Artin group has rational conjugacy growth series if and only if it is free abelian.
    • Thm: Suppose that G = GΛ is a graph product of groups G₁,...,G_n, each G_i = < X_i > with X_i finite and X_i = X_i^-1. Let X := ∪_i=1ⁿ X_i. If Geo(G_i,X_i) and ConjGeo(G_i,X_i) are regular languages for all i, then so are Geo(G,X) and ConjGeo(G,X).
    • Cor: Every right-angled Artin group and right-angled Coxeter group has rational conjugacy geodesic growth series.