Discusses such topics as the classical axiomatic systems of Euclid and Hilbert, axiom systems for three and four dimensional absolute geometry and Pieri's system based on rigid motions. Models, such as Taxicab Geometry, are used to illustrate theory.

The foundations of geometry and the non-Euclidean plane, by George E. Martin. Pp 509. $25·90. 1982. ISBN 3-540-90694-0 (Springer)

The Schreier graphs of Thompson’s group F with respect to the stabilizer of 1/2 and generators x 0 and x 1 , and of its unitary representation in L 2 ([0, 1]) induced by the standard action on the interval [0, 1] are explicitly described. The coamenability of the stabilizers of any finite set of dyadic rational numbers is established. The induced subgraph of the right Cayley graph of the positive monoid of F containing all the vertices of the form x n v, where n ≥ 0 and v is any word over the alphabet {x 0 , x 1 }, is constructed. It is proved that the latter graph is non-amenable.

Some Graphs Related to Thompson’s Group F

We study the connection between amenability, Folner conditions and the geometry of finitely generated semigroups. Using results of Klawe, we show that within an extremely broad class of semigroups (encompassing all groups, left cancellative semigroups, finite semigroups, compact topological semigroups, inverse semigroups, regular semigroups, commutative semigroups and semigroups with a left, right or two-sided zero element), left amenability coincides with the strong Folner condition. Within the same class, we show that a finitely generated semigroup of subexponential growth is left amenable if and only if it is left reversible. We show that the (weak) Folner condition is a left quasi-isometry invariant of finitely generated semigroups, and hence that left amenability is a left quasi-isometry invariant of left cancellative semigroups. We also give a new characterisation of the strong Folner condition in terms of the existence of weak Folner sets satisfying a local injectivity condition on the relevant translation action of the semigroup.

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Amenability and geometry of semigroups

David Hilbert. David Hilbert's lectures on the foundations of geometry, 1891–1902. Michael Hallett and Ulrich Majer, eds. David Hilbert's Foundational Lectures; 1. Berlin: Springer-Verlag, 2004. ISBN 3-540-64373-7. Pp. xxviii + 661

Ruinous subsets of Richard Thompson’s group F

Let T be a subsemigroup of a cancellative, left amenable semigroup S. We give conditions which are both necessary and sufficient for the subsemigroup T to be left amenable.

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Necessary and Sufficient Conditions for a Subsemigroup of a Cancellative Left Amenable Semigroup to be Left Amenable

We show that Richard Thompson's group F is nonamenable if and only if the three element set A 2 = {x 0 x 0 x 0, x 0 x 0 x 1, x 0 x 0 x 2} is weakly-ruinous. We then show that no two element set is weakly-ruinous, thereby showing that A 2 is minimal in both the number of elements that a weakly-ruinous set must contain and also in the number of carets that each tree in a weakly-ruinous set must have.

A Minimal Candidate for the Existence of a Weakly-Ruinous Set

Let M be a cancellative monoid such that the monoid ring ℤM has no zero divisors. We show that if the monoid consisting of all elements of ℤM with strictly positive coefficients has nonzero common right multiples, then M is left amenable.

A partial converse to a theorem of Tamari

Following the approach of G.D. Birkhoff, we develop a new axiom system for a continuous absolute plane. In this new axiom system we remove Side-Angle-Side as an axiom and replace it with Side-Angle-Angle as a new axiom. We prove that the new axiom system is also a continuous absolute plane, and in particular, that Side-Angle-Side holds in the new axiom system. In addition, we give new proofs of well known results. These new proofs do not depend on Side-Angle-Side, but instead use Side-Angle-Angle.

John Donnelly

Papers

Ruinous subsets of Richard Thompson’s group F

Necessary and Sufficient Conditions for a Subsemigroup of a Cancellative Left Amenable Semigroup to be Left Amenable

A Minimal Candidate for the Existence of a Weakly-Ruinous Set

A partial converse to a theorem of Tamari

The equivalence of Side-Angle-Side and Side-Angle-Angle in the absolute plane