### Group actions and Helly's theoremGroup actions and Helly's theorem

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 Author Farb, Benson Source arXiv.org Content type Text File Format PDF Date of Submission 2008-06-10 Language English
 Subject Domain (in DDC) Natural sciences & mathematics ♦ Mathematics Subject Keyword Mathematics - Group Theory ♦ Mathematics - Representation Theory ♦ math Abstract 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 $\SL_n(\Z[x_1,..., x_d]), n>2$). Educational Use Research Learning Resource Type Article Page Count 17