Disks in The Heisenberg Group - First-Year Interest Group Seminar | L A 1, Study notes of Humanities

Download Disks in The Heisenberg Group - First-Year Interest Group Seminar | L A 1 and more Study notes Humanities in PDF only on Docsity! DISKS IN THE HEISENBERG GROUP ERIC KATERMAN 1. Outline (1) Motivation (a) History (i) Dehn (ii) Cannon (iii) (W.) Thurston – formalize the notion of “seeing” the Cayley graph (b) Automatic Groups (i) fix notation: G f.g, G =< S|R >,A = S ∪ S−1, · : A∗ → G (ii) Definition I: machines (iii) Definition II: pictures (iv) easy examples (v) other examples (vi) closure properties (vii) non-examples (viii) Q: nilpotent groups? (2) Lemmas (a) short representatives: proof by machines (definition I) (b) small disks: proof by pictures (definition II) (3) Heisenberg group (a) H =< α, β, γ|[α, β]γ−1, [α, γ], [β, γ] > (b) G =< α(x, y, z) = (x+1, y, z +y), β(x, y, z) = (x, y +1, z), γ(x, y, z) = (x, y, z + 1) > implies that H ∼= H = R3/G (c) use G acting on Z3 ∈ R3 to construct Cayley graph of H in R3 (d) show slides of empty Γ, with w1, w2 (e) in general, yz-area of disk D spanning wn has area close to n 3 (f) elementary spanning surfaces correspond to conjugates of relators (i) area of Si (and its conjugates) ≤ 1, so need at least n 3 of them to fill D (ii) so minimal φ(n) is at least cubic, therefore not automatic by [2] (iii) note: combinatorial argument implies that minimal φ(n) is cubic 2. Motivation Much of this section is adapted from [Far92]. 2.1. History. In the 1910s, Dehn solved the word problem for surface groups using the geometry of H2. In 1984, Cannon extended this to cocompact discrete groups of hyperbolic isometries. The main thrust of Cannon’s paper is that one can “see” the Cayley graphs for such groups, i.e. there is a way to take a picture of the graph in 1 2 ERIC KATERMAN a finite ball around 0 and a finite machine that dictates how to stick copies of that finite piece together. Example: Γ{1}(Z). To formalize this notion, use finite-state automata (FSA). Definition 1. Let G be a finitely generated group with generating set S and let A = S ∪ S−1 = {a1, . . . , an}. We have a map A ∗ → G, denoted by w 7→ w. G is an automatic group if (1) there is an FSA M over A such that π : L → G is onto; (2) the following languages are regular: • L= = {(u, v) : u, v ∈ L(M) and u = v} • La1 = {(u, v) : u, v ∈ L(M) and u = va1} ... • • Lan = {(u, v) : u, v ∈ L(M) and u = van} This definition is equivalent to the following: G is an automatic group if (1) there is an FSA M such that π : L(M) → G is onto; (2) there is a constant k such that u, v ∈ L(M), dw(u, v) = 1 implies that u and v satisfy the k-fellow traveller property. Easy examples: finite groups, Z, F2 (drawn on side board) Other examples: hyperbolic groups, abelian groups, braid group, mapping class group Closure properties: direct product, free product, free product with amalgama- tion over a finite subgroup, finite index subgroups and supergroups, quasiconvex subgroups Non-examples: SLn(Z), n ≥ 3, non-silly Baumslag-Solitar groups Question: nilpotent groups? 3. Lemmas Lemma 3.1 (short representatives). Let G be automatic with automatic structure (A,L). There there is a constant N such that any w ∈ A∗ is equivalent to some w′ ∈ L where |w′| ≤ N |w|+ n0 where n0 is the length of an accepted representative of 1. proof by machines. First we claim that there is a constant M such that if w ∈ L, g ∈ G with dw(w, g) ≤ 1, then there is a representative u ∈ L with u = g such that |u| ≤ |w|+ M . To see this, choose M bigger than the number of states in any FSA in definition I. Let u ∈ L such that u = g, so (w, u) or (u,w) is accepted by one of the Mx for some x ∈ A∪ ². If |u| > |w|+ M , then Mx undergoes more than M transitions after reading w, so we could shorten u by removing a loop from the path of transitions in Mx. Now we take N = M and induct on |w|. If |w| = 0, w is the empty word and so it represents the identity and thus w ∼ w′ ∈ L with w′ ∈ L and w′ = n0 = N |w|+n0. If |w| > 0, then w = ux for some x ∈ A and |u| < |w|. Apply IH to u so there is a u′ ∼ u with u′ ∈ L and |u′| ≤ N(|w| − 1) + n0. But the claim implies that there is a word w′ ∈ L with w′ = w and |w′| ≤ |u′| + N ≤ N |w| + n0. ¤