Generalisations of hyperbolic groups and their properties

双曲群的概念是由Gromov在1987年提出的，灵感来自于双曲几何、低维拓扑和组合群论。双曲群的概念已经成为几何群论中的核心概念，双曲群的许多性质都是由双曲群满足的。这在小抵消理论和随机群密度模型中都得到了一些令人振奋的结果。近年来，人们对双曲群的推广进行了大量的研究。这样的推广导致了相对双曲群的概念，例如，非直线双曲群或CAT(0)群。寻找一个不是CAT(0)群的双曲群是一个众所周知的公开问题，关于非单线性双曲群已经做了大量的工作，其中群的作用是在Gromov双曲群上作用的适当不连续性的弱化。通过它在曲线复形上的作用，这类群包括双曲群、相对双曲群以及OUT(FN)和映射类群。无线性双曲群有几个有趣的性质。例如，它们是SQ-泛的，这意味着每个可数群都可以嵌入到它的一个商群中。另一个有趣的性质是西斯托博士证明的：对于有限生成的非线性双曲群G，在G上的一个简单的随机游动产生一个长度为n的“广义斜率”元素的概率随着n趋于无穷大而以指数速度收敛到1。我的博士研究将着眼于双曲群的这样或那样的推广，以及这些类群具有什么性质。

The notion of hyperbolic group was introduced in 1987 by Gromov, with inspiration from hyperbolic geometry and low dimensional topology and combinatorial group theory. The notion of a hyperbolic group has become central in geometric group theory with many properties being satisfied by hyperbolic groups. This has led to some exciting results in small cancellation theory as well as in density models for random groups. In recent years, a lot of generalisations of hyperbolic groups have been studied. Such generalisations lead to the notion of relative hyperbolic groups, acylindrically hyperbolic groups or CAT(0) groups for example. It is a well known open problem to find a hyperbolic group that is not a CAT(0) group.A lot of work has been done on acylindrically hyperbolic groups, where the action of the group is a weakening of the proper discontinuity of the action on a Gromov hyperbolic group. This class of groups includes hyperbolic groups, relative hyperbolic groups as well as Out(Fn) and the Mapping class group via its action on curve complexes. Acylindrically hyperbolic groups have several interesting properties. For example, they are SQ-universal meaning every countable group can be embedded in one of its quotient groups. Another interesting property is one proved by Dr. Sisto : for a finitely generated acylindrically hyperbolic group G, the probability that a simple random walk on G of length n produces a 'generalized loxodromic' element converges to 1 exponentially fast as n tends to infinity.My PhD research will be looking at such, and other, generalisations of hyperbolic groups and what properties these classes of groups have.