Geometric genus is an important invariant in the classification theory for isolated singularities. In this paper we give a complete classification of three-dimensional isolated weighted homogeneous singularities with geometric genus one. This is one of important classes of minimally elliptic singularities.

For any r-order {0,1}-tensor A with e ones, we prove that the spectral radius of A is at most exp((r-1)/r) with the equality holds if and only if e=k^r for some integer k and all ones forms a principal sub-tensor 1_{k x k x ... x k}. We also prove a stability result for general tensor A with e ones where e = k^r+l with relatively small l. Using the stability result, we completely characterized the tensors achieving the maximum spectral radius among all r-order {0,1}-tensor A with k^r+l ones, for -r-1 ≤ l ≤ r, and k sufficiently large.

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We construct small cancellation labellings for some infinite sequences of finite graphs of bounded degree. We use them to define infinite graphical small cancellation presentations of groups. This technique allows us to provide examples of groups with exotic properties: • We construct the first examples of finitely generated coarsely non-amenable groups (that is, groups without Guoliang Yu’s Property A) that are coarsely embeddable into a Hilbert space. Moreover, our groups act properly on CAT(0) cubical complexes. • We construct the first examples of finitely generated groups, with expanders embedded isometrically into their Cayley graphs—in contrast, in the case of the Gromov monster expanders are not even coarsely embedded. We present further applications.

We associate to a finite digraph D a lattice polytope PD whose vertices are the rows of the Laplacian matrix of D. This generalizes a construction introduced by Braun and the third author. As a consequence of the Matrix-Tree Theorem, we show that the normalized volume of PD equals the complexity of D, and PD contains the origin in its relative interior if and only if D is strongly connected. Interesting connections with other families of simplices are established and then used to describe reflexivity, the h∗-polynomial, and the integer decomposition property of PD in these cases. We extend Braun and Meyer’s study of cycles by considering cycle digraphs. In this setting, we characterize reflexivity and show there are only four non-trivial reflexive Laplacian simplices having the integer decomposition property.