Amenability via random walks

By Laurent Bartholdi, B'alint Vir'ag. 2003

168 citations recorded by the source corpus.

We use random walks to show that the Basilica group is amenable, answering an open question of Grigorchuk and ˙ Zuk. Our results separate the class of amenable groups from the closure of subexponentially growing groups under the operations of group extension and direct limits; these classes are separated even within the realm of finitely presented groups.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7eb7aa10617697203ae6cbe401f6874fa4e83676

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

On simplicity of reduced C*-algebras of groups (2005, background): “For other examples, see [32], [54], and [8].”
Extensions of amenable groups by recurrent groupoids (2013, methodology): “tructed from groups of sub-exponential growth (which, in some sense, can be also considered as an “easy case” of amenability) is the basilica group introduced in [GZ02]. Its amenability was proved in [BV05] using asymptotic propertie˙ s of random walks on groups. These methods were then generalized ”
On amenability of automata groups (2008, methodology): “If the group G is self-similar, then, as it was explained in Section 4.A above, any random walk (G, � ) gives rise to a RWIDF (G × X, M � ). In [ BV05 ] and [Kai05] one passed then from the RWIDF (G × X, M � ) to a new random walk (G, � ′) by taking the trace of the RWIDF (G × X, M � ) on a single “”
Poisson-Furstenberg Boundaries, Large-scale Geometry and Growth of Groups (2011, background): “A very interesting class of examples was discovered recently by Bartholdi and Virag [11], who studied a group that was defined earlier by Grigorchuk and Żuk.”
Group Laws and Free Subgroups in Topological Groups (2003, background): “([Wi1]), Segal’s group of slow subgroup growth ([Seg]) and the group constructed by Grigorchuk and Zuk which is not in the class SG ([GZ]) but as Bartholdi and Vir´ag have recently shown is amenable ([BV]). A proﬁnite weakly branch group is a weakly branch group which is closed in the proﬁnite topol”
Classification of groups generated by 3-state automata over a 2-letter alphabet (2008, background): “ew interesting groups and an attempt to use them to solve known problems. An example of such a group is the Basilica group (see automaton [852]). It is the ﬁrst example of an amenable group (shown in [BV05]) that is not sub-exponentially amenable group (shown in [GZ02a]).˙ 2. Recognition of already ”
Amenability of linear-activity automaton groups (2009, methodology): “Ascension of random walks first considered in Bartholdi and Virág (2005).”
Milnor's Problem on the Growth of Groups and its Consequences (2011, methodology): “The Munchhausen trick has been used to prove amenability of certain self-similar groups of exponential growth [25, 23, 5].”
Schreier graphs of the Basilica group (2009, background): “Virág further showed it to be amenable, making Basilica the first example of an amenable but not subexponentially amenable group [3].”