Publications


  1. Amenability and geometry of semigroups (with M. Kambites)
    Transactions of the American Mathematical Society (to appear)
  2. On regularity and the word problem for free idempotent generated semigroups (with I. Dolinka and N. Ruskuc)
    Proceedings of the London Mathematical Society Vol. 114, 2017, pp. 401-432.
  3. Diagram monoids and Graham-Houghton graphs: idempotents and generating sets of ideals (with J. East)
    Journal of Combinatorial Theory, Series A Vol. 146, 2017, pp. 63-128.
  4. Motzkin monoids and partial Brauer monoids (with I. Dolinka and J. East)
    Journal of Algebra Vol. 471, 2017, pp. 251-298.
  5. Automorphism groups of countable algebraically closed graphs and endomorphisms of the random graph (with I. Dolinka, J. D. McPhee, J. D. Mitchell, and M. Quick)
    Mathematical Proceedings of the Cambridge Philosophical Society Vol. 160, 2016, pp. 437-462.
  6. Ends of Semigroups (with S. Craik, V. Kilibarda, J. D. Mitchell, and N. Ruskuc)
    Semigroup Forum Vol. 93, 2016, pp. 330-346.
  7. Rewriting systems and biautomatic structures for Chinese, hypoplactic, and Sylvester monoids (with A. J. Cain and A. Malheiro)
    International Journal of Algebra and Computation Vol. 25, 2015, no. 1-2.
  8. A strong geometric hyperbolicity property for directed graphs and monoids (with M. Kambites )
    Journal of Algebra Vol. 420, 2014, pp. 373-401.
  9. The minimal number of generators of a finite semigroup
    Semigroup Forum Vol. 89, 2014, pp. 135-154.
  10. Homotopy Bases and Finite Derivation Type for Subgroups of Monoids (with A. Malheiro)
    Journal of Algebra Vol. 410, 2014, pp. 53-84.
  11. Countable locally 2-arc-transitive bipartite graphs (with J. K. Truss)
    European Journal of Combinatorics Vol. 39, 2014, pp. 122-147.
  12. Maximal subgroups of free idempotent generated semigroups over the full linear monoid (with I. Dolinka)
    Transactions of the American Mathematical Society Vol. 366, 2014, pp. 419-455.
  13. Ideals and finiteness conditions for subsemigroups (with V. Maltcev, J. D. Mitchell and N. Ruskuc)
    Glasgow Mathematical Journal Vol. 56, 2014, pp. 65-86.
  14. Quasi-isometry and finite presentations for left cancellative monoids (with M. Kambites)
    International Journal of Algebra and Computation Vol. 23, 2013, pp. 1099-1114.
  15. Groups Acting on Semimetric Spaces and Quasi-isometries of Monoids (with M. Kambites)
    Transactions of the American Mathematical Society Vol. 365, 2013, pp. 555-578.
  16. Homotopy bases and finite derivation type for Schützenberger groups of monoids (with A. Malheiro and S. J. Pride)
    Journal of Symbolic Computation Vol. 50, 2013, pp. 50-78.
  17. Maximal subgroups of free idempotent generated semigroups over the full transformation monoid (with N. Ruskuc)
    Proceedings of the London Mathematical Society Vol. 104, 2012, pp. 997-1018.
  18. Set-homogeneous Directed Graphs (with D. Macpherson, C. Praeger and G. Royle)
    Journal of Combinatorial Theory, Series B Vol. 102, 2012, pp. 474--520.
  19. On Maximal Subgroups of Free Idempotent Generated Semigroups (with N. Ruskuc)
    Israel Journal of Mathematics Vol. 189, 2012, pp. 147-176.
  20. A Svarc-Milnor Lemma for Monoids Acting by Isometric Embeddings (with M. Kambites)
    International Journal of Algebra and Computation Vol. 21, 2011, pp. 1135-1147.
  21. Presentations of Inverse Semigroups their Kernels and Extensions (with C. Carvalho and N. Ruskuc)
    Journal of the Australian Mathematical Society Vol. 90, 2011, pp. 289-316.
  22. Homological Finiteness Properties of Monoids, their Ideals and Maximal Subgroups (with S. J. Pride)
    Journal of Pure and Applied Algebra Vol. 215, 2011, pp. 3005-3024.
  23. On properties not inherited by monoids from their Schützenberger groups (with A. Malheiro and S. J. Pride)
    Information and Computation Vol. 209, 2011, pp. 1120-1134.
  24. Generators and Relations for Subsemigroups via Boundaries in Cayley Graphs (with N. Ruskuc)
    Journal of Pure and Applied Algebra Vol. 215, 2011, pp. 2761-2779.
  25. Locally-finite Connected-homogeneous Digraphs (with R. Moller)
    Discrete Mathematics Vol. 311, 2011, pp. 1497-1517.
  26. Finite Complete Rewriting Systems for Regular Semigroups (with A. Malheiro)
    Theoretical Computer Science Vol. 412, 2011, pp. 654-661.
  27. Countable connected-homogeneous graphs (with D. Macpherson)
    Journal of Combinatorial Theory, Series B Vol. 100, 2010, pp. 97-118.
  28. Cycle-free Partial Orders and Ends of Graphs (with J. K. Truss)
    Mathematical Proceedings of the Cambridge Philosophical Society Vol. 146, 2009, pp. 535-550.
  29. On Residual Finiteness of Direct Products of Algebraic Systems (with N. Ruskuc)
    Monatshefte fur Mathematik Vol. 158, 2009, pp. 63-69.
  30. k-CS-transitive Infinite Graphs
    Journal of Combinatorial Theory, Series B Vol. 99, 2009, pp. 378-398.
  31. Green Index and Finiteness Conditions for Semigroups (with N. Ruskuc)
    Journal of Algebra Vol. 320, 2008, pp. 3145-3164.
  32. Construction of Some Countable One-arc Transitive Bipartite Graphs (with J. K. Truss)
    Discrete Mathematics Vol. 308, 2008, pp. 6392-6405.
  33. Hall's Condition and Idempotent Rank of Ideals of Endomorphism Monoids
    Proceedings of the Edinburgh Mathematical Society Vol. 51, 2008, pp. 1-16.
  34. Largest Subsemigroups of the Full Transformation Monoid (with J. D. Mitchell)
    Discrete Mathematics Vol. 308, 2008, pp. 4801-4810.
  35. Idempotent Rank in Endomorphism Monoids of Finite Independence Algebras
    Proceedings of the Royal Society of Edinburgh: Section A Mathematics Vol. 137A, 2007, pp. 303-331.
  36. Generating Sets of Completely 0-Simple Semigroups (with N. Ruskuc)
    Communications in Algebra Vol. 33, 2005, pp. 4657-4678.
Submitted papers
Thesis
[Preprint from arXiv] [Final published journal version]
Abstract
We study the connection between amenability, Følner 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 Følner 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) Følner 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 Følner condition, in terms of the existence of weak Følner sets satisfying a local injectivity condition on the relevant translation action of the semigroup.