Title: Uniform exponential growth alternative for CAT(0) cubical groups with IFP
Speaker: Thomas Ng (Temple University)
Abstract: A group is said to have uniform exponential growth if the number of elements that can be spelled with words of bounded length is bounded below by a single exponential function over all generating sets. In 1981, Gromov asked whether all groups with exponential growing group in fact have uniform exponential growth. While this was shown not to be the case in general, it has been answered affirmatively for many natural classes of groups such as hyperbolic groups, linear groups, and the mapping class groups of a surface. In 2018, Kar-Sageev show that groups acting properly on 2-dimensional CAT(0) cube complexes by loxodromic isometries either have uniform exponential growth or are virtually abelian by explicitly exhibiting free semigroups whose generators have uniformly bounded word length whenever they exist. These free semigroups witness the uniform exponential growth. I will describe how certain arrangements of hyperplane orbits can be used to build loxodromic isometries and show that they generate free semigroups in the generalized context of CAT(0) cubical groups with the isolated flats property in arbitrary dimension.
This is joint work with Radhika Gupta and Kasia Jankiewicz.
