We give a solution to the inverse problem (given a prescribed function, find a corresponding group) for large classes of speed, entropy, isoperimetric profile, return probability and Lp-compression functions of finitely generated groups. For smaller classes, we give solutions among solvable groups of exponential volume growth. As corollaries, we prove a recent conjecture of Amir on joint evaluation of speed and entropy exponents and we obtain a new proof of the existence of uncountably many pairwise non-quasi-isometric solvable groups, originally due to Cornulier and Tessera. We also obtain a formula relating the Lp-compression exponent of a group and its wreath product with the cyclic group forp in [1,2].

No abstract uploaded!

We prove that for any free ergodic probability measure-preserving action $${\mathbb{F}_n \curvearrowright (X, \mu)}$$ of a free group on$n$generators $${\mathbb{F}_n, 2\leq n \leq \infty}$$ , the associated group measure space II_{1}factor $${L^\infty (X)\rtimes \mathbb{F}_n}$$ has$L$^{∞}($X$) as its unique Cartan subalgebra, up to unitary conjugacy. We deduce that group measure space II_{1}factors arising from actions of free groups with different number of generators are never isomorphic. We actually prove unique Cartan decomposition results for II_{1}factors arising from arbitrary actions of a much larger family of groups, including all free products of amenable groups and their direct products.

No abstract uploaded!

Multivariate cryptography is one of the main candidates to guarantee the security of communication in the presence of quantum computers. While there exist a large number of secure and efficient multivariate signature schemes, the number of practical multivariate encryption schemes is somewhat limited. In this paper we present our results on creating a new multivariate encryption scheme, which is an extension of the original SimpleMatrix encryption scheme of PQCrypto 2013. Our scheme allows fast en- and decryption and resists all known attacks against multivariate cryptosystems. Furthermore, we present a new idea to solve the decryption failure problem of the original SimpleMatrix encryption scheme.