# Generalized triangle groups, expanders, and a problem of Agol and Wise

## TL;DR

This paper demonstrates that a stronger version of Wise's malnormal special quotient theorem does not hold, using counterexamples based on generalized triangle groups and Ramanujan graphs.

## Contribution

It provides the first counterexamples to a stronger form of Wise's theorem, constructed via generalized triangle groups and Ramanujan graphs.

## Key findings

- Stronger Wise's theorem does not hold.
- Counterexamples are based on generalized triangle groups.
- Uses Ramanujan graphs for construction.

## Abstract

Answering a question asked by Agol and Wise, we show that a desired stronger form of Wise's malnormal special quotient theorem does not hold. The counterexamples are generalizations of triangle groups, built using the Ramanujan graphs constructed by Lubotzky--Phillips--Sarnak.