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. 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, Cayley Graph, the Word 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 group in 1854. I In 1878, Cayley wrote four papers on group theory. 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 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, 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 Poul Heegaard, first comprehensive article on topology (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 Ruth Moufang. 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 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, Dorothea Rockburne. 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-manifolds. 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 = fx1; :::; xng be a finite set of group generators. Define the −1 −1 −1 set X = fx1 ; :::; xn g 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 free group 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 = f! 2 F j ! = ρi ri ρi ; with n 2 Z, ρi 2 F , ri 2 Rg: i=1 Definition: G = hX jRi 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 j s ; r ; srsri, Z2 ∗ Z3 = hx; y j x ; y i, Z × Z = ha; b j 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 f(g; a; ga) j g 2 G; a 2 A = X[X −1g. 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 j s ; r ; srsri, Z2 ∗ Z3 = hx; y j x ; y i, Z × Z = ha; b j 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 j Ri, a solution to the word problem for G is an algorithm that can take any word ! 2 A∗ and determine, in a finite number of steps, whether or not the word ! is equivalent to the identity in G. Definition: The Conjugacy Problem Given G = hX j Ri, a solution to the conjugacy problem for G is an algorithm that can take any two words !; ν 2 A∗ and determine, in a finite number of steps, whether or not the word ! is conjugate to ν in G (i.e. 9 g 2 G such that ! = 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 j abABi. Note that w = [a(abAB)A][aba(abAB)ABA][ab(abAB)BA][(abAB)] in the free group on fa; bg. 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 ! = 1 in G = hX j 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 j!j 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. Dehn Function Definition: Isoperimetric Inequality. Let jwj 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 2 F with w = 1, n ≤ f (jwj). 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 fn required for all w 2 F such that w = 1 and jwj ≤ xg: 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.