from max dehn to mikhael gromov, the geometry of infinite ...goddard/mini/2011/peifer.pdf ·...

From Max Dehn to Mikhael Gromov,the Geometry 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.


Outline

I Groups and Geometry before 1900

I Max Dehn

I Infinite Groups – Presentations, Cayley Graph, the WordProblem

I Geometry of Groups – Dehn Diagrams, IsoparametricInequality

I Dehn’s Algorithm

I Mikhael Gromov

I Hyperbolic groups


Groups before 1900

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

I Lagrange in 1770, and Cauchy in 1844, had studiedpermutations.

I Abel and Galois

I Arthur Cayley gave the first abstract definition of a group in1854.

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

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


Geometry before 1900

Modern Geometry

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

I Bernhard Riemann (1854)– analytic approach

I David Hilbert (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, Poincare – the study of surfaces


Poincare Disk Model of H2

A

B

C

Figure: Poincare’s hyperbolic disk model of H2, with two geodesictriangles. One triangle is finite while the other is ideal, meaning itsvertices lie at points of infinity.


Max Dehn (1878-1952) - early career

I Doctorate (1900) Univ of Gottingen under David Hilbert.

I Habilitation (1901) Univ of Munster. Solved Hilbert’s thirdproblem: Archimedean postulate is logically needed to provethat tetrahedra of equal base and height have equal volume.

I 1907, co-authored, with Poul Heegaard, first comprehensivearticle on topology (analysis situs).

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

I Taught at Univ of Munster, Kiel, and Breslau

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

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

I Students: Wilheim Magnus and Ruth Moufang.


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 Instituteof Technology.

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

I Last seven years of his life at Black Mountain College (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, DorotheaRockburne.


Max Dehn ’s 1910-1914 papers

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

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

I Found an early example of a Poincare homology sphere.Introduces Dehn surgery, still important in the study of3-manifolds.

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


Infinite Groups and Presentations

Let X = {x1, ..., xn} be a finite set of group generators. Define theset X−1 = {x−11 , ..., x−1n } and the set of semigroup generatorsA = X ∪ X−1.

Definition: A∗ denotes the set of all words of finite length on A.This is the semigroup on A.

Words xix−1i and x−1i xi are called inverse pairs.

A word in A∗ is said to be reduced if it contains no inverse pairsub-words.

Definition: F denotes the subset of A∗ of all reduced words. F isisomorphic to the free group on X .

The product of two words in F is the result of concatenating thewords and deleting inverse pairs. The empty word is the identity.


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 asthe smallest normal subgroup of F containing the elements of R.

Fact: N = {ω ∈ F | ω =n∏

i=1

ρi riρ−1i , with n ∈ Z, ρi ∈ F , ri ∈ R}.

Definition: G = 〈X |R〉 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.

Examples: D3 = 〈s, r | s3, r2, srsr〉, Z2 ∗ Z3 = 〈x , y | x2, y3〉,Z× Z = 〈a, b | abAB〉, and F2 = 〈p, q〉.


Cayley Graphs

Definition: Cayley graphLet 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 groupelements.

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

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

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


Figure: Cayley graphs: D3 = 〈s, r | s3, r2, srsr〉, Z2 ∗ Z3 = 〈x , y | x2, y3〉,Z× Z = 〈a, b | abAB〉, and F2 = 〈p, q〉.


Decision Problems (Dehn 1912)

Definition: The Word Problem

Given G = 〈X | R〉, a solution to the word problem for G is analgorithm that can take any word ω ∈ A∗ and determine, in afinite number of steps, whether or not the word ω is equivalent tothe identity in G .

Definition: The Conjugacy Problem

Given G = 〈X | R〉, a solution to the conjugacy problem for G isan algorithm that can take any two words ω, ν ∈ A∗ anddetermine, in a finite number of steps, whether or not the word ωis conjugate to ν in G (i.e. ∃ g ∈ G such that ω = gνg−1).


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Figure: w = a2babA2BAB = 1 in Z× Z = 〈a, b | abAB〉. Note thatw = [a(abAB)A][aba(abAB)ABA][ab(abAB)BA][(abAB)] in the freegroup on {a, b}.


Dehn Diagrams and Area

Intuitive Definition: Dehn diagram (Van-Kampen diagram, andsmall cancellation diagram)

A Dehn diagram for a word ω = 1 in G = 〈X | R〉 is a simplyconnected 2-dimensional diagram made up of regions whichintersect at edges. The edges of the diagram are labeled withsubwords of relators, such that reading around any interior regionis a relator and the word w labels the boundary of the diagram.

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


Dehn Function

Definition: Isoperimetric Inequality.Let |w | represent the length of a word w in A∗ (or in F ). A groupis 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 takesas 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.


Surface Groups

Theorem: (Dehn, Heegaard) The surfaces (closed orientable2-manifolds without boundary) are homeomorphic to the sphere ora “torus” with one or more holes. The number of holes is calledthe genus, g , of the surface.

Definition: The fundamental group of a surface, π1(S), is called asurface group.

Theorem: (Poincare) A surfaces of genus g can be represented bya 4g -gon, with a specific gluing pattern on the edges. These4g -gons can tesselate a hyperbolic plane of constant curvature,while still obeying the gluing pattern. The elements of the surfacegroup act on the hyperbolic plane by translations that map the netof 4g -gons onto itself. The surface group acts freely on thishyperbolic tesselation by translations. The hyperbolic plane is theuniversal cover for this group action.


Fundamental Polygons

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


Figure: Covering of genus 2 surface by octagon tiling of H2.


Dehn’s Algorithm

Suppose that G = 〈X | R〉 is a surface group, ω = 1 in G , and ωis not the empty word.

Using Cayley graphs and hyperbolic geometry, Dehn showed thatany Dehn diagram for ω must contain at least one region with thefollowing property. More than half of the region’s boundary liesalong the boundary of the diagram. Thus more than half of arelator is included in the word ω.

Solution to the word problem: Examine the word ω to determinewhether or not it has a subword that is more than half a relator. Ifso, replace the subword by the shorter half of the relator and checkagain. 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 islinear in the length of ω.


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 ofMathematics, in France. Soviets did not allow him to attend.Submitted an influential paper on differential equations to theconference proceedings.

During 1979, gave lectures at the Universite de Paris VII oncurvature of a Riemannian manifold and its global behavior.

In 1981, position at the Universite de Paris VI. In 1982 moved tothe Institut des Hautes Etudes Scientifiques, where he is now apermanent member.

US positions; SUNY Stony Brook, Univ of Maryland, College Park,the Courant Institute of Mathematical Sciences, and Cornell Univ.


Mikhael Gromov

In 2009, Gromov was awarded the prestigious Abel Prize. A quotefrom this award reads, “The Russian-French mathematicianMikhail L Gromov is one of the leading mathematicians of ourtime. ... Mikhail Gromov has led some of the most importantdevelopments, producing profoundly original general ideas whichhave resulted in new perspectives on geometry and other areas ofmathematics. Gromov’s name is forever attached to deep resultsand important concepts within Riemannian geometry, symplecticgeometry, string theory and group theory”.


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 newgeometric prospective for studying infinite groups given bypresentations.

Definition: A geodesic triangle is δ-thin if given any point x on oneside, there is a path to a point y on one of the other sides, withlength less than or equal to δ.

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Figure: A δ-thin triangle.


Gromov’s Hyperbolic groups

Definition: A finitely generated group G = 〈X | R〉 is hyperbolic ifthere exists a δ ≥ 0 such that all geodesic triangles in the Cayleygraph of G are δ-thin.

Theorem: (Gromov) A finitely generated group G = 〈X | R〉 ishyperbolic iff is has a linear Dehn function.

Fact: Using Teitze transformations it can be shown that thepolynomial degree of the Dehn function for a group is independentof the presentation.


Free groups are hyperbolic

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


Z× Z is not hyperbolic

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


The Geometry of Infinite Groups


Fellow Travelers

Definition: Fellow TravelersTwo paths w and v 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.

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Definition: BicombingLet G = 〈X |R〉 be a group with A = X ∪ X−1. A languageL ⊂ 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 ∪ ∅ andw , v ∈ L with aw = vb in G , then the paths w and v in theCayley graph ΓA(G ) are δ-fellow travelers.

Definition: Biautomatic GroupA group G = 〈X |R〉 is biautomatic if there is a language L ⊂ A∗that is regular and a bicombing of G .


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


Biautomatic Groups

Theorem: Biautomatic groups satisfy a quadratic isoperimetricinequality.

Theorem: Biautomatic groups have a solvable conjugacy problem.

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