From Max Dehn to Mikhael Gromov, the of Infinite Groups.

Dave Peifer

University of North Carolina at Asheville

October 28, 2011

Dave Peifer From Max Dehn to Mikhael Gromov, the Geometry of Infinite Groups. The Geometry of Infinite Groups

Figure: Max Dehn.

Dave Peifer From Max Dehn to Mikhael Gromov, the Geometry of Infinite Groups. Outline

I Groups and Geometry before 1900

I Max Dehn

I Infinite Groups – Presentations, , the Problem

I Geometry of Groups – Dehn Diagrams, Isoparametric Inequality

I Dehn’s Algorithm

I Mikhael Gromov

I Hyperbolic groups

Dave Peifer From Max Dehn to Mikhael Gromov, the Geometry of Infinite Groups. Groups before 1900

I Euler in 1761, and later Gauss in 1801, had studied modular arithmetic.

I Lagrange in 1770, and Cauchy in 1844, had studied permutations.

I Abel and Galois

I Arthur Cayley gave the first abstract definition of a in 1854.

I In 1878, Cayley wrote four papers on . Introduced a combinatorial graph associated to the group given by a set of generators.

I Walther Von Dyck, in 1882 and 1883, introduced a way to present a group in terms of generators and relations.

Dave Peifer From Max Dehn to Mikhael Gromov, the Geometry of Infinite Groups. Geometry before 1900

Modern Geometry

I Discovery of Hyperbolic Geometry. Bolyai (1832), Lobachevsky (1830), Gauss

I Bernhard Riemann (1854)– analytic approach

I (1899) – axiomatic approach Connections – Geometry and Groups

I In 1872, Felix Klein outlined his Enlangen Program.

I In 1884, Sophus Lie began studying Lie groups.

I In 1895 and 1904, Poincar´e– the study of surfaces

Dave Peifer From Max Dehn to Mikhael Gromov, the Geometry of Infinite Groups. Poincar´eDisk Model of H2

B

A

C

Figure: Poincar´e’shyperbolic disk model of H2, with two geodesic triangles. One triangle is finite while the other is ideal, meaning its vertices lie at points of infinity.

Dave Peifer From Max Dehn to Mikhael Gromov, the Geometry of Infinite Groups. Max Dehn (1878-1952) - early career

I Doctorate (1900) Univ of G¨ottingenunder David Hilbert.

I Habilitation (1901) Univ of M¨unster.Solved Hilbert’s third problem: Archimedean postulate is logically needed to prove that tetrahedra of equal base and height have equal volume.

I 1907, co-authored, with , first comprehensive article on (analysis situs).

I 1910-1914 published a series of papers on topology and infinite groups.

I Taught at Univ of M¨unster,Kiel, and Breslau

I 1921-1935, Chair (Ordinarius) at the University of Frankfurt.

I History Seminar (Carl Siegel, analytic number theorist). Dehn knew several languages (including Greek and Latin), he was a naturalist, and he loved and studied music and the arts.

I Students: Wilheim Magnus and .

Dave Peifer From Max Dehn to Mikhael Gromov, the Geometry of Infinite Groups. Max Dehn - later career

I 1938, arrested the day after Kristallnacht.

I Escape to Norway and eventually to the United States.

I US positions Univ of Idaho, in Pocatello, and Illinois Institute of Technology.

I One year at St John’s College, in Annapolis, MD.

I Last seven years of his life at (BMC), in North Carolina.

I BMC – Bauhaus Art School, Avant-garde of modern art. Included Joseph Albers, Jack Tworkov, Merce Cunningham, Kenneth Snelson, John Cage, Buckmister Fuller, .

Dave Peifer From Max Dehn to Mikhael Gromov, the Geometry of Infinite Groups. Max Dehn ’s 1910-1914 papers

I Presented the word problem, conjugacy problem, and isomorphism problem for groups given by presentations.

I Uses hyperbolic geometry to solve the word and conjugacy problems for the surface groups.

I Found an early example of a Poincar´ehomology sphere. Introduces Dehn surgery, still important in the study of 3-.

I Studied the fundamental group of the complement to the trefoil knot. By solving the word problem for this group, proved that the right and left trefoil knots are not homotopic.

Dave Peifer From Max Dehn to Mikhael Gromov, the Geometry of Infinite Groups. Infinite Groups and Presentations

Let X = {x1, ..., xn} be a finite set of group generators. Define the −1 −1 −1 set X = {x1 , ..., xn } and the set of semigroup generators A = X ∪ X −1. Definition: A∗ denotes the set of all words of finite length on A. This is the semigroup on A. −1 −1 Words xi xi and xi xi are called inverse pairs. A word in A∗ is said to be reduced if it contains no inverse pair sub-words. Definition: F denotes the subset of A∗ of all reduced words. F is isomorphic to the on X . The product of two words in F is the result of concatenating the words and deleting inverse pairs. The empty word is the identity.

Dave Peifer From Max Dehn to Mikhael Gromov, the Geometry of Infinite Groups. Infinite Groups and Presentations

Let R be a set of reduced words in F . Definition: The normal closure of R, denoted by N, is defined as the smallest normal subgroup of F containing the elements of R. n Y −1 Fact: N = {ω ∈ F | ω = ρi ri ρi , with n ∈ Z, ρi ∈ F , ri ∈ R}. i=1 Definition: G = hX |Ri signifies that G is isomorphic to F /N, where F is the free group on X , and N is the normal closure of R. Fact: Every group has a presentation.

3 2 2 3 Examples: D3 = hs, r | s , r , srsri, Z2 ∗ Z3 = hx, y | x , y i, Z × Z = ha, b | abABi, and F2 = hp, qi.

Dave Peifer From Max Dehn to Mikhael Gromov, the Geometry of Infinite Groups. Cayley Graphs

Definition: Cayley graph Let G be a group with generating set X , the Cayley graph of G, denoted by ΓA(G), has vertex set G and edge set {(g, a, ga) | g ∈ G, a ∈ A = X ∪ X −1}. Notice that

I ΓA(G) is a directed graph, I The vertex set is in 1-1 correspondence with the group elements.

I The edges represent multiplication on the right by the generator or inverse generator.

I The structure of ΓA(G) looks exactly the same starting at any vertex.

I ΓA(G) is a geodesic metric space with the word metric.

Dave Peifer From Max Dehn to Mikhael Gromov, the Geometry of Infinite Groups. 3 2 2 3 Figure: Cayley graphs: D3 = hs, r | s , r , srsri, Z2 ∗ Z3 = hx, y | x , y i, Z × Z = ha, b | abABi, and F2 = hp, qi. Dave Peifer From Max Dehn to Mikhael Gromov, the Geometry of Infinite Groups. Decision Problems (Dehn 1912)

Definition: The Word Problem Given G = hX | Ri, a solution to the word problem for G is an algorithm that can take any word ω ∈ A∗ and determine, in a finite number of steps, whether or not the word ω is equivalent to the identity in G.

Definition: The Conjugacy Problem Given G = hX | Ri, a solution to the conjugacy problem for G is an algorithm that can take any two words ω, ν ∈ A∗ and determine, in a finite number of steps, whether or not the word ω is conjugate to ν in G (i.e. ∃ g ∈ G such that ω = gνg −1).

Dave Peifer From Max Dehn to Mikhael Gromov, the Geometry of Infinite Groups. a

b b

a a a a b b b b b b a a a b a b a a b a a a a b b a

a b a b a b a

a a b b a b

a a a a

b b b b b b

a a a a a a

b b b b b

a a a a b a

a b

Figure: w = a2babA2BAB = 1 in Z × Z = ha, b | abABi. Note that w = [a(abAB)A][aba(abAB)ABA][ab(abAB)BA][(abAB)] in the free group on {a, b}.

Dave Peifer From Max Dehn to Mikhael Gromov, the Geometry of Infinite Groups. Dehn Diagrams and Area

Intuitive Definition: Dehn diagram (Van-Kampen diagram, and small cancellation diagram) A Dehn diagram for a word ω = 1 in G = hX | Ri is a simply connected 2-dimensional diagram made up of regions which intersect at edges. The edges of the diagram are labeled with subwords of relators, such that reading around any interior region is a relator and the word w labels the boundary of the diagram.

Notice that the number n of relators needed in the algebraic product is the number of regions required in the Dehn diagram. While |ω| is measuring the length of the boundary, n is measuring the number of regions, or the area, enclosed by w.

Dave Peifer From Max Dehn to Mikhael Gromov, the Geometry of Infinite Groups.

Definition: Isoperimetric Inequality. Let |w| represent the length of a word w in A∗ (or in F ). A group is said to have a isoperimetric inequality if there is a function f (x) such that for all words w ∈ F with w = 1, n ≤ f (|w|).

Definition: Dehn Fuction. Notice that there is a well defined function f : N → N that takes as input a given length x and returns the maximum of the set

{n required for all w ∈ F such that w = 1 and |w| ≤ x}.

This function is called a Dehn function for the presentation.

Dave Peifer From Max Dehn to Mikhael Gromov, the Geometry of Infinite Groups. Surface Groups

Theorem: (Dehn, Heegaard) The surfaces (closed orientable 2-manifolds without boundary) are homeomorphic to the sphere or a “torus” with one or more holes. The number of holes is called the genus, g, of the surface.

Definition: The fundamental group of a surface, π1(S), is called a surface group. Theorem: (Poincar´e)A surfaces of genus g can be represented by a 4g-gon, with a specific gluing pattern on the edges. These 4g-gons can tesselate a hyperbolic of constant curvature, while still obeying the gluing pattern. The elements of the surface group act on the hyperbolic plane by translations that map the net of 4g-gons onto itself. The surface group acts freely on this hyperbolic tesselation by translations. The hyperbolic plane is the universal cover for this group action.

Dave Peifer From Max Dehn to Mikhael Gromov, the Geometry of Infinite Groups. Fundamental Polygons

Figure: The polygons for surfaces of genus 1 and 2.

Dave Peifer From Max Dehn to Mikhael Gromov, the Geometry of Infinite Groups. Figure: Covering of genus 2 surface by octagon tiling of H2.

Dave Peifer From Max Dehn to Mikhael Gromov, the Geometry of Infinite Groups. Dehn’s Algorithm

Suppose that G = hX | Ri is a surface group, ω = 1 in G, and ω is not the empty word. Using Cayley graphs and hyperbolic geometry, Dehn showed that any Dehn diagram for ω must contain at least one region with the following property. More than half of the region’s boundary lies along the boundary of the diagram. Thus more than half of a relator is included in the word ω. Solution to the word problem: Examine the word ω to determine whether or not it has a subword that is more than half a relator. If so, replace the subword by the shorter half of the relator and check again. If the word reduces to the empty word, then ω = 1. If not, then ω 6= 1. Note: Assuming G is finitely presented, then Dehn’s algorithm is linear in the length of ω.

Dave Peifer From Max Dehn to Mikhael Gromov, the Geometry of Infinite Groups. Mikhael Gromov

Born December 23, 1943 in Boksitogorsk, Russia. Doctorate in 1969 from Leningrad University. Professor at Leningrad University, 1967-1974. In 1970, invited to speak at the International Congress of , in France. Soviets did not allow him to attend. Submitted an influential paper on differential equations to the conference proceedings. During 1979, gave lectures at the Universit´ede Paris VII on curvature of a Riemannian and its global behavior. In 1981, position at the Universit´ede Paris VI. In 1982 moved to the Institut des Hautes Etudes´ Scientifiques, where he is now a permanent member. US positions; SUNY Stony Brook, Univ of Maryland, College Park, the Courant Institute of Mathematical Sciences, and Cornell Univ.

Dave Peifer From Max Dehn to Mikhael Gromov, the Geometry of Infinite Groups. Mikhael Gromov

In 2009, Gromov was awarded the prestigious Abel Prize. A quote from this award reads, “The Russian-French mathematician Mikhail L Gromov is one of the leading mathematicians of our time. ... Mikhail Gromov has led some of the most important developments, producing profoundly original general ideas which have resulted in new perspectives on geometry and other areas of mathematics. Gromov’s name is forever attached to deep results and important concepts within , symplectic geometry, string theory and group theory”.

Dave Peifer From Max Dehn to Mikhael Gromov, the Geometry of Infinite Groups. Gromov’s Hyperbolic groups

Gromov wrote a series of papers on infinite groups in the 1980’s. In On Hyperbolic Groups (1987), Gromov explores his new geometric prospective for studying infinite groups given by presentations. Definition: A geodesic triangle is δ-thin if given any x on one side, there is a path to a point y on one of the other sides, with length less than or equal to δ.

q x ≤ δ q y q q

q Figure:A δ-thin triangle.

Dave Peifer From Max Dehn to Mikhael Gromov, the Geometry of Infinite Groups. Gromov’s Hyperbolic groups

Definition: A finitely generated group G = hX | Ri is hyperbolic if there exists a δ ≥ 0 such that all geodesic triangles in the Cayley graph of G are δ-thin. Theorem: (Gromov) A finitely generated group G = hX | Ri is hyperbolic iff is has a linear Dehn function. Fact: Using Teitze transformations it can be shown that the polynomial degree of the Dehn function for a group is independent of the presentation.

Dave Peifer From Max Dehn to Mikhael Gromov, the Geometry of Infinite Groups. Free groups are hyperbolic

Figure: A 0-thin geodesic triangle in f F2.

Dave Peifer From Max Dehn to Mikhael Gromov, the Geometry of Infinite Groups. Z × Z is not hyperbolic

y z

t t

x

t

Figure: A 5-thin geodesic triangle in Z × Z.

Dave Peifer From Max Dehn to Mikhael Gromov, the Geometry of Infinite Groups. The Geometry of Infinite Groups

Dave Peifer From Max Dehn to Mikhael Gromov, the Geometry of Infinite Groups. Fellow Travelers

Definition: Fellow Travelers Two pathsw ˆ andv ˆ in ΓA(G) are δ-fellow travelers if for all t > 0, the distance d(w(t), v(t)) is less than or equal to δ.

wˆ (t) a ≤ δ aω = bν s s e s vˆ(t) v s s s

Dave Peifer From Max Dehn to Mikhael Gromov, the Geometry of Infinite Groups. Definition: Bicombing Let G = hX |Ri be a group with A = X ∪ X −1. A language L ⊂ A∗ is called a bicombing if it satisfies the properties below.

I L contains at least one representative for each element of G. That is, the canonical map from L to G is onto.

I There exists a constant δ such that, if a, b ∈ A ∪ ∅ and w, v ∈ L with aw = vb in G, then the pathsw ˆ andv ˆ in the Cayley graph ΓA(G) are δ-fellow travelers.

Definition: Biautomatic Group A group G = hX |Ri is biautomatic if there is a language L ⊂ A∗ that is regular and a bicombing of G.

Dave Peifer From Max Dehn to Mikhael Gromov, the Geometry of Infinite Groups. y z

u u

x

u

Figure: The language {anbm} is a bicombing of Z × Z.

Dave Peifer From Max Dehn to Mikhael Gromov, the Geometry of Infinite Groups. Biautomatic Groups

Theorem: Biautomatic groups satisfy a quadratic isoperimetric inequality. Theorem: Biautomatic groups have a solvable conjugacy problem.

µ - µ(i)

r r r  XX XX `  XXXX  ` ωj

6 6  XX XX XX`` ωi g g

ν - ν(i) r r r

Dave Peifer From Max Dehn to Mikhael Gromov, the Geometry of Infinite Groups.