{"abstracts":[{"sha1":"fb7fa41c1df5b883f73af37614982c0f0c101982","content":"Let $F$ denote the Thompson group with standard generators $A=x_0$, $B=x_1$.\nIt is a long standing open problem whether $F$ is an amenable group. By a\nresult of Kesten from 1959, amenability of $F$ is equivalent to $$(i)\\qquad\n||I+A+B||=3$$ and to $$(ii)\\qquad ||A+A^{-1}+B+B^{-1}||=4,$$ where in both\ncases the norm of an element in the group ring $\\mathbb{C} F$ is computed in\n$B(\\ell^2(F))$ via the regular representation of $F$. By extensive numerical\ncomputations, we obtain precise lower bounds for the norms in $(i)$ and $(ii)$,\nas well as good estimates of the spectral distributions of $(I+A+B)^*(I+A+B)$\nand of $A+A^{-1}+B+B^{-1}$ with respect to the tracial state $\\tau$ on the\ngroup von Neumann Algebra $L(F)$. Our computational results suggest, that\n$$||I+A+B||\\approx 2.95 \\qquad ||A+A^{-1}+B+B^{-1}||\\approx 3.87.$$ It is\nhowever hard to obtain precise upper bounds for the norms, and our methods\ncannot be used to prove non-amenability of $F$.","mimetype":"application/x-latex","lang":"en"},{"sha1":"adb4c086f9385fea7f3055d3e15c9ec3fcfab807","content":"Let F denote the Thompson group with standard generators A=x_0, B=x_1.\nIt is a long standing open problem whether F is an amenable group. By a\nresult of Kesten from 1959, amenability of F is equivalent to (i) \n||I+A+B||=3 and to (ii) ||A+A^-1+B+B^-1||=4, where in both\ncases the norm of an element in the group ring C F is computed in\nB(ℓ^2(F)) via the regular representation of F. By extensive numerical\ncomputations, we obtain precise lower bounds for the norms in (i) and (ii),\nas well as good estimates of the spectral distributions of (I+A+B)^*(I+A+B)\nand of A+A^-1+B+B^-1 with respect to the tracial state τ on the\ngroup von Neumann Algebra L(F). Our computational results suggest, that\n||I+A+B||≈ 2.95 ||A+A^-1+B+B^-1||≈ 3.87. It is\nhowever hard to obtain precise upper bounds for the norms, and our methods\ncannot be used to prove non-amenability of F.","mimetype":"text/plain","lang":"en"}],"refs":[],"contribs":[{"index":0,"raw_name":"S. Haagerup","role":"author"},{"index":1,"raw_name":"U. Haagerup","role":"author"},{"index":2,"raw_name":"M. Ramirez-Solano","role":"author"}],"license_slug":"ARXIV-1.0","language":"en","version":"v1","number":"CPH-SYM-DNRF92","ext_ids":{"arxiv":"1409.1486v1"},"release_year":2014,"release_date":"2014-09-04","release_stage":"submitted","release_type":"report","webcaptures":[],"filesets":[],"files":[{"release_ids":["2r6h526xxrcn5efubfoaay47pi"],"mimetype":"application/pdf","urls":[{"url":"https://web.archive.org/web/20200828233326/https://arxiv.org/pdf/1409.1486v1.pdf","rel":"webarchive"},{"url":"https://arxiv.org/pdf/1409.1486v1.pdf","rel":"repository"}],"sha256":"4cb39aecf5d8164ce2f9d2b3ce9665f6ed8784e00aafe04d9db42a59d93598f2","sha1":"2e6b19a5006c8f20170663226fa4423ac19a0caf","md5":"31a07b757fd191ab1eaaeb5ed667daa3","size":777235,"revision":"56634458-a074-4276-945c-395453df95ff","ident":"lxh7jkmtr5ft5in7yeoscyp4ja","state":"active"}],"work_id":"cex3icmfkvhzriipbf7zkxwmda","title":"A computational approach to the Thompson group F","state":"active","ident":"2r6h526xxrcn5efubfoaay47pi","revision":"075a939b-8b25-4941-8ef3-5523f04cce8b","extra":{"arxiv":{"base_id":"1409.1486","categories":["math.GR","math.OA"],"comments":"appears in International Journal of Algebra and Computation (2015)"},"superceded":true}}