By the density of a finite graph we mean its average vertex degree. For an m-generated group, the density of its Cayley graph in a given set of generators, is the supremum of densities taken over all its finite subgraphs. It is known that a group with m generators is amenable if and only if the density of the corresponding Cayley graph equals 2m. A famous problem on the amenability of R. Thompson’s group F is still open. Due to the result of Belk and Brown, it is known that the density of its Cayley graph in the standard set of group generators {x0,x1}, is at least 3.5. This estimate has not been exceeded so far. For the set of symmetric generators S = {x1,x̄1}, where x̄1 = x1x0−1, the same example only gave an estimate of 3. There was a conjecture that for this generating set equality holds. If so, F would be non-amenable, and the symmetric generating set would have the doubling property. This would mean that for any finite set X ⊂ F, the inequality |S±1X|≥ 2|X| holds. In this paper, we disprove this conjecture showing that the density of the Cayley graph of F in symmetric generators S strictly exceeds 3. Moreover, we show that even larger generating set S0 = {x0,x1,x̄1} does not have doubling property.

中文翻译：

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对称生成元中 R.Thompson 群 F 的 Cayley 图的密度

有限图的密度是指它的平均顶点度。为米-生成群，它的凯莱图在给定的一组生成器中的密度，是其所有有限子图的密度的上确界。据了解，一组与米当且仅当相应凯莱图的密度等于2米. 关于 R. Thompson 群的顺从性的一个著名问题F仍然开放。由于 Belk 和 Brown 的结果，已知其 Cayley 图在标准群生成器集中的密度{X0,X1}, 至少是3.5. 到目前为止，这个估计还没有被超过。对于对称生成器集小号 = {X1,X̄1}， 在哪里X̄1 = X1X0-1，同一个例子只给出了一个估计3. 有一个猜想，对于这个发电机组等式成立。如果是这样的话，F将是不适合的，并且对称发电机组将具有倍增特性。这意味着对于任何有限集X ⊂ F, 不等式|小号±1X|≥ 2|X|持有。在本文中，我们反驳了这一猜想，表明凯莱图的密度为F在对称生成器中小号严格超过3. 此外，我们展示了更大的发电机组小号0 = {X0,X1,X̄1}没有加倍属性。