Periodic Reporting for period 3 - WordMeasures (Word Measures in Groups and Random Cayley Graphs)
Reporting period: 2023-02-01 to 2024-07-31
The main theme of the project is the study of word measures in groups. This field sits in a crossroad between several areas of mathematics: combinatorial and geometric group theory, topology, combinatorics and random matrices. The basic questions involves two groups: the 'inducting' group and the 'induced' group. A fixed element of the inducting group induces a probability measure on the induced group, namely, inscribed a way to sample a random element from the induced group. For example, the inducting group may be the 'free group' consisting of formal words, and the fixed element may be the formal word abab^-1. The induced group may be the group of permutations of 100 elements. To sample a abab^-1-random permutation, we sample two uniformly random permutations on 100 elements, x and y, and consider their composition according to the template given by the word. Namely: the random permutation xyxy^-1. We then study the properties of this random permutation.
It turns out that this naive question leads to deep and fundamental mathematics. The properties of this random permutation are related to topological and algebraic 'invariants' (numerical functions) of the word in question. When instead of random permutation one considers random matrix groups such as random unitary matrices, yet other invariants of the words control the distribution of the random elements. Other invariants show up when one alters the free group with some other 'inducting' group (for example, 'surface groups').
What the study reveals time after time is that there is very deep structure in the distributions at hand. At times, the invariants controlling the distributions are previously known and the study of word measures sheds new light on them. At other times, the invariants are first discovered within the study of word measures, but than others find the play important roles in other questions in mathematics.
The goal of this project is to expose as much as possible of this structure, to understand (in a conjectural level) and then prove what the invariants controlling the distributions are, and to find a unifying theory for the many different groups we can study. In addition, we work to use these findings in different applications: in the study of random walks and spectral gaps and in the study of profinite topology of groups.
It turns out that this naive question leads to deep and fundamental mathematics. The properties of this random permutation are related to topological and algebraic 'invariants' (numerical functions) of the word in question. When instead of random permutation one considers random matrix groups such as random unitary matrices, yet other invariants of the words control the distribution of the random elements. Other invariants show up when one alters the free group with some other 'inducting' group (for example, 'surface groups').
What the study reveals time after time is that there is very deep structure in the distributions at hand. At times, the invariants controlling the distributions are previously known and the study of word measures sheds new light on them. At other times, the invariants are first discovered within the study of word measures, but than others find the play important roles in other questions in mathematics.
The goal of this project is to expose as much as possible of this structure, to understand (in a conjectural level) and then prove what the invariants controlling the distributions are, and to find a unifying theory for the many different groups we can study. In addition, we work to use these findings in different applications: in the study of random walks and spectral gaps and in the study of profinite topology of groups.
In the first period of the project we have achieved the following:
- We established a much more elaborated conjectural picture for word measures on various groups, and, to some extent, a conjectural unifying structure. For many groups we established partial results towards this conjectural picture.
- For an important class of groups (linear matrix groups over finite fields) we established an intriguing connection between word measures on these groups and free-group algebras.
- Together with Michael Magee and other co-authors, we studied random permutations with probability distributions induced by non-free groups, such as surface groups and free products of finite groups. We found beautiful and sometimes surprising structure here, and also described more detailed conjectures.
- Together with Michael Magee and other co-authors, we used our results on random permutations sampled by surface groups to establish new results about the spectral gap of the Laplacian operator on random hyperbolic surfaces.
- We established a much more elaborated conjectural picture for word measures on various groups, and, to some extent, a conjectural unifying structure. For many groups we established partial results towards this conjectural picture.
- For an important class of groups (linear matrix groups over finite fields) we established an intriguing connection between word measures on these groups and free-group algebras.
- Together with Michael Magee and other co-authors, we studied random permutations with probability distributions induced by non-free groups, such as surface groups and free products of finite groups. We found beautiful and sometimes surprising structure here, and also described more detailed conjectures.
- Together with Michael Magee and other co-authors, we used our results on random permutations sampled by surface groups to establish new results about the spectral gap of the Laplacian operator on random hyperbolic surfaces.
We have by now numerous conjectures about probability measures induced on groups by other 'inducting' groups. My hope is that by the end of the project, we will be able to prove many of these conjectures, to clarify the conjectural picture (based on proofs but also simulations and concrete examples), and to use our results to establish some fundamental problems about random Cayley and Schreier graphs.