Group actions and Helly's theorem

By Benson Farb. 2008

53 citations recorded by the source corpus.

We describe a connection between the combinatorics of generators for certain groups and the combinatorics of Helly's 1913 theorem on convex sets. We use this connection to prove fixed point theorems for actions of these groups on nonpositively curved metric spaces. These results are encode d in a property that we introduce called ``property $\FA_r$'', which reduces to Serre's property $\FA$ when $r=1$. The method applies to $S$-arithmetic groups in higher $\Q$-rank, to simplex reflection groups (including some non-arithmetic ones), and to higher rank Chevalley groups over polynomial and other rings (for example $\SL_n(\Z[x_1,..., x_d]), n>2$).

Subfields: Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/c12a9df9aae11f54042b7e06a8922ba6c630cd61

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In Computational Theory and Mathematics

In Geometry and Topology

Citations in the corpus

Groups acting on manifolds: around the Zimmer program (2008, background): “be described in terms of representations of the ambient groups, e.g. SL(n,R), and will be determined by specifying a numerical value of X. Some partial results towards superrigidity are known by Farb [51] and Shenﬁeld [212]. The problem of completely classifying linear representations in this contex”
On the joint spectral radius for isometries of non-positively curved spaces and uniform growth (2018, unknown): “oof for -hyperbolic spaces is a quasication of the corresponding proof for isometries of trees. It involves in particular a Helly-type theorem for hyperbolic spaces, which we prove in Section6. See [39] and [28] for a related use of Helly-type theorems in geometric group theory. As already mentioned”
On parabolic subgroups of Artin groups (2022, unknown): “However, if H is not ﬁnitely generated, the set { Fix(Φ( h )) | h ∈ H } , which does consist of closed and convex subsets, can in general not be written as a ﬁnite intersection hence the result of [Far09, Thm. 3.2] is not applicable.”
Isometry groups of non‐positively curved spaces: structure theory (2008, background): “A fixed-point property for similar groups acting on low-dimensional CAT(0) cell complexes was established by Farb [34].”