| Week |
Date of Monday |
Topics |
Lecture notes |
Example sheet |
Comments |
| 1 |
Jan. 6 |
Admin. Motivation. Groups, examples, and standard
exercises. Homomorphisms, isomorphsims, automorphisms. New
groups from old: subgroups, direct products, semi-direct
products. NIL is isomorphic to the Heisenburg group.
Automorphism groups. Group actions. Words, length, empty word,
letters, concatenation, subwords, prefixes, suffixes, powers.
Inverses, reductions, expansions. Reduced word theorem and the
diamond lemma (peak reduction). Free groups, cancellation
lemma. Many exercises.
|
Tuesday
Wednesday
Thursday
|
One
|
Tuesday: A student asks "What is the universal property of the
semi-direct product?"
Thursday: A student points out that the boards are difficult to
read on the lecture capture from the Wednesday lecture.
|
| 2 |
Jan. 13 |
Universal property of free groups. Generators, finitely
generated groups, rank. Normal closures, normally generating.
Presentations, every group has a presentation, presentations are
never unique. Finitely presented groups. Relations versus
relators. Many presentations of the trivial group.
Non-finitely generated groups. Cayley graphs, groups act on
their Cayley graphs via graph automorphisms. Edge paths, edge
loops, labels. The Cayley graph of \(F(S)\) is a tree.
|
Tuesday
Wednesday
Thursday
|
Two
|
|
| 3 |
Jan. 20 |
Universal property of finite presentations. Word problem,
independence from generating set. The word problem is hard.
Edge metric, word metric, word norm. Groups act on their Cayley
graphs via isometries. Computing from a finite presentation.
Different sets of generators give comparable word metrics.
Undistorted and distorted subgroups. Geodesics, geodesic words,
short lex geodesic words. Short lex representatives, the short
lex spanning tree. Volume growth, for free abelian groups, for
free groups.
|
Tuesday
Wednesday
Thursday
|
Three
|
Tuesday: A student asks: "What can be computed from a
finite presentation?"
|
| 4 |
Jan. 27 |
Growth rate is (essentially) invariant under change of
generators. Growth characterisation of \(\mathbb{Z}^n\).
Sublinear and superexponential growth are impossible. Nilpotent
groups. Theorems of Hirsch, Jennings, Bass-Guivarc'h, and Gromov
- polynomial growth if and only if virtually nilpotent (and
intra-polynomial growth is impossible). Virtual properties.
Theorem of Wilkie-van den Dries - linear growth implies
virtually \(\mathbb{Z}\). Geodesic rays and lines. Ends of
groups. Ends appear eventually. Ends never die.
|
Tuesday
Wednesday
Thursday
|
Four
|
|
| 5 |
Feb. 3 |
Approximate ends, diameter. Ends never die, take two. Number
of ends is well-defined, is independent of generating set. The
set of ends. Approximate ends contain geodesic rays. The
number of ends equals the cardinality of the set of ends. The
group acts on its ends. A group has two ends if and only if it
is virtually infinite cyclic. Boundaries, separators.
|
Tuesday
Wednesday
Thursday
|
Five
|
Note that I got the definition of "end-equivalence" slightly
wrong in lecture on Wednesday. I fixed this in-lecture on
Thursday (and in both sets of lecture notes).
|
| 6 |
Feb. 10 |
Neighbourhoods, geodesic metric spaces. Geodesic triangles,
slim triangles, hyperbolic metric spaces. Hyperbolic groups.
Examples. Slim quadrilaterals. Non-examples. Local geodesics.
Local geodesics (of sufficient quality) embed in hyperbolic
spaces. Non-torsion elements exist. Torsion elements are
small.
|
Tuesday
Wednesday
Thursday
|
Six
|
Wednesday: A student asks "Is hyperbolicity (of a finitely
presented group) a decidable property?" The answer is "no"
because (as we will hopefully prove) \(\mathbb{Z}^2\) never
embeds in a hyperbolic group.
Wednesday: The diagram I was trying to draw in class (a
geodesic rectangle in the Cayley graph of \(BS(1, 2)\)) is
drawn more neatly in the revised lecture notes.
Thursday: A question I asked in class: is there a finitely
presented group that has infinitely many conjugacy classes of
torsion elements? The answer is "yes" (for example,
Houghton's group \(H_3\)), but I do not know of an elementary
discussion.
|
| 7 |
Feb. 17 |
Outradius and outcentres. Geodesic, complete, proper metric
spaces. The diameter of the set of outcentres is uniformly
bounded in a hyperbolic space. Controling finite subgroups of
hyperbolic groups. Local geodesics are quasi-geodesics.
Hyperbolic groups are finitely presented, have Dehn
presentations. Surface groups (genus at least two) have a Dehn
presentation. Hyperbolic groups have solvable word problem.
The conjugacy problem, begun.
|
Tuesday
Wednesday
Thursday
|
Seven
|
Tuesday: Here is a version of a question I asked in class:
Suppose that \(\Gamma\) is a connected graph equipped with the
edge metric. Suppose that \(K \geq 0\) and \(\Gamma\) has the
following property: for every finite non-empty set \(\Delta
\subset \Gamma\), the set of outcentres of \(\Delta\) has
diameter at most \(K\). Is \(\Gamma\) \(\delta\)-hyperbolic for
some \(\delta\)?
|
| 8 |
Feb. 24 |
The conjugacy problem, finished. Quasi-convexity, finite
generation, and non-distortion. Centralisers in hyperbolic
groups are quasi-convex. Quasi-isometric embeddings,
quasi-surjective funstions, and quasi-isometries. Sharp
triangles. Divergence of geodesics. Quasi-geodesics. Morse
stability, begun.
|
Tuesday
Wednesday
Thursday
|
Eight
|
Wednesday: A student asks "Is being a Dehn presentation a
decidable property (of a finite presentation)?" This appears
to be an open question, previously asked by Delzant. For
closely related work, see the
following paper
of Arzhantseva.
|
| 9 |
Mar. 3 |
Morse stability, finished. Quasi-isometry invariants:
hyperbolicity, number of ends, growth, virtual nilpotence,
virtual commutativity, finite presentability. Geometric
actions, Milnor-Svarc lemma (the fundamental observation of
geometric group theory). Cyclic subgroups (in hyperbolic
groups) are undistorted, are virtually cyclic.
|
Tuesday
Wednesday
Thursday
|
Nine
|
|
| 10 |
Mar. 10 |
Axes. Short-lex geodesics fellow-travelling quasi-geodesics.
Uniformly rational translation length. Closest points
projection (to geodesics). Almost-centralisers. Ping-pong
lemma. Free subgroups of hyperbolic groups. Gromov boundaries.
|
Tuesday
Wednesday
Thursday
|
Ten
|
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