#Simon thomas minimal subshift simple amenable full

#Simon thomas minimal subshift simple amenable full

It also shows amenabil-ity of the topological full groups of a class of non-minimal subshifts with slow complexity (seeSubsection 1.2). Our approach does not rely on results in and yields a new proof of amenability of the groups that we consider. They arise as the commutator subgroups of the topological fullgroups of some minimal subshifts, on which we assume that the complexity grows slowly enough(these notions are deﬁned in Subsection 1.1). Theorem 1.1 is a consequence of Theorem 1.2 below.The groups that we consider to prove Theorem 1.1 are a sub-class of the ﬁnitely gener-ated simple groups discovered by Matui and considered byJuschenko and Monod. Moreover, there areuncountably many pairwise non-isomorphic such groups.

There exist ﬁnitely generated inﬁnite groups that are simple and have the Liou-ville property (for every symmetric, ﬁnitely supported probability measure).

For a recent survey on Poisson-Furstenbergboundaries of random walks on discrete groups, see. However, there are ﬁnitely generated amenable groups, suchas the wreath product Z / Z ≀ Z, that admit no ﬁnitely supported, non-degenerate measureswith trivial boundary, see Kaimanovich and Vershik on some amenablegroups, a non-degenerate measure with trivial boundary might not even be chosen to have ﬁniteentropy by a result of Erschler. More precisely, a group is amenable if, and only if, it admits a symmetric non-degenerate measure µ with trivial Poisson-Furstenberg boundary (one implication is due to ∗ Universit´e Paris Sud urstenberg, see, the other to Kaimanovich and Vershik and to Rosenblatt ). Finitely generated groups with sub-exponential growth havethe Liouville property (this is due to Avez ), and groups with the Liouville propertyare amenable. When no measure is speciﬁed, we say that the group G has the Liouville property if ( G, µ )has the Liouville property for every symmetric, ﬁnitely supported probability measure µ on G ,including degenerate measures. If the support of µ generates G the measure is said to be non-degenerate. Here a function f : G → R is said tobe µ -harmonic if f ∗ µ = f, where ( f ∗ µ )( g ) = P h ∈ G f ( gh ) µ ( h ). G, µ ) has the Liouville property if the Poisson-Furstenberg boundary is trivial equivalently, if every bounded µ -harmonic function on G isconstant on the subgroup generated by the support of µ.

We prove thatthere exist simple groups with the Liouville property.A group equipped with a probability measure ( We consider here a third property of groups that lies between sub-exponential growth andamenability: the Liouville property for ﬁnite-range symmetric random walks. Recently, Juschenko and Monod have proven that there do exist ﬁnitelygenerated, simple groups that are amenable the groups that they consider were known to besimple and ﬁnitely generated by results of Matui. Recall that groups of sub-exponential growthare amenable. It is considerably less understood how“small” can such groups be, from the point of view of their asymptotic geometry.It follows from Gromov’s theorem that a ﬁnitely generated simple group can nothave polynomial growth, and it is an open question, due to Grigorchuk (see ), whether it can have sub-exponential growth. Thus, ﬁnitely generatedsimple groups can be arbitrarily “large” in some sense.

Later, Hall, Gorjuˇskin, and Schupp showed that anycountable group can be embedded in a 2-generated simple group. In the early 50s Graham Higman gave the ﬁrst example of a ﬁnitely generated, inﬁnite simplegroup. We also get explicit upper bounds for thegrowth of Følner sets. We show thatif the (not necessarily minimal) subshift has a complexity function that grows slowly enough (e.g.linearly), then every symmetric and ﬁnitely supported probability measure on the topologicalfull group has trivial Poisson-Furstenberg boundary. Results by Matui andJuschenko-Monod have shown that the derived subgroups of topological full groups of minimalsubshifts provide the ﬁrst examples of ﬁnitely generated, simple amenable groups. We study random walk on topological full groups of subshifts, and show the existence ofinﬁnite, ﬁnitely generated, simple groups with the Liouville property. M a y Subshifts with slow complexity and simple groups with theLiouville property