A counterexample to Sorenson’s conjecture: The finitely generated case

Yuji, Takahashi

doi:10.1007/s00233-001-0170-y

Cited by 3 publications

(3 citation statements)

References 8 publications

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“…For finitely generated semigroups in absolute generality, it is well known that left FC does not imply left amenability ( [16], or see [34,Section 4.22] for a more accessible reference) and that left amenability does not imply left SFC (due to an example of [38] which is a refinement to the finitely generated case of a result of Klawe [29]). We have seen that the existence of an element of maximal local cogrowth does not imply left amenability (see Remark 6.16 above), and therefore nor does maximal global cogrowth, or the associated random walk Markov operator having a norm or spectral radius of at least 1.…”

Section: Argabright and Wilde [1]mentioning

confidence: 99%

“…The focus so far has been chiefly on the finite case (most notably on face monoids of hyperplane arrangements), but the extension to finitely generated semigroups is arguably the next natural step. Finally, amenability for semigroups has been an active area of research from the 1950s [12] to the present day; see [2,3,4,10,23] for some examples of recent developments and applications in other areas and [18,38] for advances relating specifically to finitely generated semigroups. It therefore seems apposite to ask whether whether the concept of cogrowth makes sense in a semigroup setting, and if so whether it is capable of forming a similar bridge between these areas.…”

Section: Introductionmentioning

confidence: 99%

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On Cogrowth, Amenability, and the Spectral Radius of a Random Walk on a Semigroup

Gray

Kambites

2018

International Mathematics Research Notices

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We introduce two natural notions of cogrowth for finitely generated semigroups -one local and one global -and study their relationship with amenability and random walks. We establish the minimal and maximal possible values for cogrowth rates, and show that non-monogenic free semigroups are exactly characterised by minimal global cogrowth. We consider the relationship with cogrowth for groups and with amenability of semigroups. We also study the relationship with random walks on finitely generated semigroups, and in particular the spectral radius of the associated Markov operators (when defined) on ℓ2-spaces. We show that either of maximal global cogrowth or the weak Følner condition suffices for its spectral radius to be at least 1; since left amenability implies the weak Følner condition, this represents a generalisation to semigroups of one implication of Kesten's Theorem for groups. By combining with known results about amenability, we are able to establish a number of new sufficient conditions for (left or right) amenability in broad classes of semigroups. In particular, maximal local cogrowth left implies amenability in any left reversible semigroup, while maximal global cogrowth (which is a much weaker property) suffices for left amenability in an extremely broad class of semigroups encompassing all inverse semigroups, left reversible left cancellative semigroups and left reversible regular semigroups.2000 Mathematics Subject Classification. 05C81, 20M05, 20M10, 60B15, 60J10.

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Section: Argabright and Wilde [1]mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On Cogrowth, Amenability, and the Spectral Radius of a Random Walk on a Semigroup

Gray

Kambites

2018

International Mathematics Research Notices

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“…The stronger form ("the strong Følner condition" or SFC ) was considered by Argabright and Wilde [1]; they showed that SFC is sufficient for left amenability, and that necessity would follow from a conjecture of Sorenson [18,19], asserting that every right cancellative, left amenable semigroup was also left cancellative. Subsequently, Klawe [15] showed that necessity was actually equivalent to Sorenson's conjecture, before disproving both, by producing a (non-finitely-generated) example of a left amenable semigroup which was right but not left cancellative; this was subsequently refined to a finitely generated example by Takahashi [20]. Despite considerable further work in this area (see for example [23]), an elementary combinatorial characterisation of (left or two-sided) amenability for semigroups remains elusive.…”

Section: Introductionmentioning

confidence: 99%

Amenability and geometry of semigroups

Gray¹,

Kambites²

2015

Preprint

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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.

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Amenability and geometry of semigroups

Gray

Kambites

2017

Trans. Amer. Math. Soc.

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A counterexample to Sorenson’s conjecture: The finitely generated case

Cited by 3 publications

References 8 publications

On Cogrowth, Amenability, and the Spectral Radius of a Random Walk on a Semigroup

On Cogrowth, Amenability, and the Spectral Radius of a Random Walk on a Semigroup

Amenability and geometry of semigroups

Amenability and geometry of semigroups

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