Introduction to Thompson’s group Ken Brown Cornell University Abstract Forty years ago Richard Thompson introduced a fascinating discrete group F , which has become a test case for many questions in geometric group theory. I will describe F from several different points of view and state some known results and open problems. 1
1. Thompson’s definition F < T < V F is the group of associative laws, T allows cyclic rearrangements, V allows arbitrary rearrangements. We only consider F . x 0 : a ( bc ) → ( ab ) c x 1 : a ( b ( cd )) → a (( bc ) d ) x 2 : a ( b ( c ( de ))) → a ( b (( cd ) e )) Expansion: Replace a, b, c, . . . by expressions. A ( BC ) → ( AB ) C A ( B ( CD )) → A (( BC ) D ) a ( b ( c ( de ))) → ( ab )( c ( de )) ( ab )( c ( de )) → ( ab )(( cd ) e ) 2
Composition x 0 a ( b ( c ( de ))) ( ab )( c ( de )) x 1 x 0 x 2 x 1 a ( b (( cd ) e )) ( ab )(( cd ) e ) x 0 A relation: x x 0 x 1 x 0 = x 0 x 2 1 = x 2 or More generally, x x i x n x i = x i x n +1 n = x n +1 ( i < n ) or Fact: x 0 , x 1 , x 2 , . . . generate F , and these are defining relations. 3
2. Combinatorial group theory F = � x 0 , x 1 , x 2 , . . . ; x x i n = x n +1 for i < n � x n x i → x i x n +1 (smaller subscripts first) x − 1 i x n → x n +1 x − 1 (positive before negative) i n x i → x i x − 1 x − 1 (positive before negative) n +1 x − 1 → x − 1 n +1 x − 1 i x − 1 (smaller subscripts last) n i Normal forms: f = x i 1 x i 2 · · · x i k x − 1 j l · · · x − 1 j 2 x − 1 ( i 1 ≤ · · · ≤ i k , j 1 ≤ · · · ≤ j l ) j 1 Unique if reduced : If x i and x − 1 both occur, then so does x i +1 or x − 1 i +1 . i x 0 x 1 x 1 x 3 x − 1 5 x − 1 x − 1 1 x − 1 = x 0 x 1 x 2 x − 1 4 x − 1 x − 1 4 0 3 0 4
3. Group of fractions F is the group of right fractions of its positive semigroup P : ⇒ f = pq − 1 f ∈ F = ( p, q ∈ P ) P has a concrete interpretation as the semigroup of binary forests (Belk, Brin). 5
x 0 · · · x 1 · · · x 2 · · · 6
x 0 · · · x 1 · · · x 0 x 1 · · · 7
Relations x 1 x 0 = x 0 x 2 · · · x 2 x 0 = x 0 x 3 · · · 8
4. Dyadic PL-homeomorphisms of I (or R + or R ) F ∼ = PL 2 ( I ) [ ∼ = PL 2 ( R + ) ∼ = PL 2 ( R ) ]. All slopes are integral powers of 2 , all breakpoints have dyadic rational coordinates, integer translation near ±∞ if use R + or R . 9
PL 2 ( I ) ∼ = PL 2 ( R + ) ∼ = PL 2 ( R ) 0 10
5. Tree and forest diagrams Binary trees encode binary subdivisions or parenthesized expressions. If use R + , get forest diagrams (but we knew this already). If use R , get doubly-infinite forest diagrams. 11
6. Universal conjugacy idempotent (Freyd–Heller, Dydak) F is the universal example of a group with an endomorphism that is idempotent up to conjugacy: φ 2 = φ x 0 φ ( x n ) = x n +1 , Homeomorphism interpretation: φ ( f ) = “ f concentrated on [1 / 2 , 1] ”. Universality: Given any φ : G → G with φ 2 conjugate to φ , need x 0 so that φ 2 = φ x 0 , (1) then need x 1 = φ ( x 0 ) , x 2 = φ ( x 1 ) ,. . . . Equation (1) forces x n +1 = x x 0 ( n > 0) , n apply φ to get remaining relations. 12
7. Algebra automorphisms (Galvin–Thompson) F is isomorphic to the group of order-preserving automorphisms of a free Cantor algebra: µ : X × X → X (bijection) Everything splits uniquely as a product. a = a 0 a 1 = a 0 ( a 10 a 11 ) a = a 0 a 1 = ( a 00 a 01 ) a 1 Every tree diagram (or associative law) gives an automorphism. a a x 0 a 0 a 1 a 0 a 1 a 10 a 11 a 00 a 01 13
Why is F interesting? • Comes up in many ways. • Has interesting properties. • Almost every question is a challenge. 14
Known properties of F 1. Good finiteness properties: Two generators x 0 , x 1 . Two relations x x 0 x 0 = x x 0 x 1 and x x 0 x 0 x 0 = x x 0 x 0 x 1 . And so on (Brown–Geoghegan). 1 1 1 1 2. F is “almost simple”. 1 → F ′ → F → Z × Z → 0 3. Although highly nonabelian, F admits a product F × F → F , associative up to conjugacy. [No identity; 1 ∗ 1 = 1 , but 1 ∗ f = φ ( f ) in general.] 4. F has no free subgroups. 5. F is not elementary amenable. 6. Isoperimetric constant with respect to x 0 , x 1 is ≤ 1 / 2 (Belk–Brown). 7. The Poisson boundary for (some) symmetric random walks is nontrivial (Kaimanovich). 8. Homology is known (B–G): H n ( F ) ∼ = Z ⊕ Z for all n ≥ 1 . 15
9. Homology and cohomology are known as rings (B): H ∗ ( F ) is an associative algebra (without identity) generated by e (degree 0), α, β (degree 1), subject to relations e 2 = e eα = βe = 0 αe = α , eβ = β Consequence: α 2 = β 2 = 0 , alternating products αβα · · · and βαβ · · · give basis in positive dimensions. H ∗ ( F ) ∼ � ( a, b ) ⊗ Γ( u ) . = 10. F is orderable. 11. Easy algorithm for computing length function (Fordham, Belk–Brown). 12. Growth series explicitly known for P (Burillo, B–B): 1 − x 2 p ( x ) = 1 − 2 x − x 2 + x 3 16
Open problems 1. Is F amenable? 2. Is the isoperimetric constant 1 / 2 ? Is it ≥ 1 / 2 for all generating sets? 3. Describe the Poisson boundary or other invariants of random walk. 4. Is F automatic? 5. What is the exponential growth rate of F ? 6. Is the growth series rational? 7. Is it true that every subgroup of F is either elementary amenable or contains an isomorphic copy of F ? See http://www.aimath.org/WWN/ for more (to appear). 17
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