Geometric group theory

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Required prerequisites, all the way down: 50

You will learn to

• Draw Cayley graphs, compute word metrics and decide when two groups are quasi-isometric.
• State and apply the Švarc–Milnor lemma to relate fundamental groups to universal covers.
• Define δ-hyperbolic groups and amenable groups, and describe Gromov's polynomial growth theorem.

Definition

• The Cayley graph of a group G with finite generating set S has the elements of G as vertices and an edge from g to gs for each s ∈ S.
• A map f : X → Y between metric spaces is a quasi-isometry if there are constants with (1/λ) d(x, y) − c ≤ d(f(x), f(y)) ≤ λ d(x, y) + c and every point of Y lies within c of the image.
• A geodesic metric space is δ-hyperbolic if every side of every geodesic triangle lies within δ of the union of the other two sides.

Theorem

• Švarc–Milnor: if G acts properly and cocompactly by isometries on a proper geodesic space X, then G is finitely generated and quasi-isometric to X.
• Gromov: a finitely generated group has polynomial growth exactly when it has a nilpotent subgroup of finite index.
• Fundamental groups of closed manifolds of negative sectional curvature are hyperbolic groups.
• Banach–Tarski: assuming the axiom of choice, a solid ball in ℝ³ can be decomposed into finitely many pieces that reassemble into two balls of the same size.

Common misconceptions

• Thinking the Cayley graph depends only on the group; it depends on the generating set, though its quasi-isometry class does not.
• Believing Banach–Tarski contradicts measure theory; the pieces are non-measurable.

Optional discoveries

• The axiom of choice and its equivalents The Banach–Tarski paradox uses a free subgroup of SO(3) together with the axiom of choice; non-amenable groups are exactly those admitting such paradoxical decompositions.
• Random structures: walks, percolation and random geometry Random walks on a group see its geometry: Kesten showed a finitely generated group is amenable exactly when the return probability decays subexponentially.
• ← Groups
• ← Riemannian geometry
• ← Low-dimensional topology: 3- and 4-manifolds
• ← Random structures: walks, percolation and random geometry

Further reading

• Matt Clay and Dan Margalit (editors), Office Hours with a Geometric Group Theorist (2017)
• Martin R. Bridson and André Haefliger, Metric Spaces of Non-Positive Curvature (1999)
• Clara Löh, Geometric Group Theory: An Introduction (2017)